Method and apparatus for complex phase equalization for use in a communication system

ABSTRACT

Method and apparatus for complex phase equalization in a communication system. The method includes receiving and downconverting a two-dimensional communication signal to generate in-phase (I) and quadrature (Q) components that have been phase-rotated by an unknown amount during propagation from a transmitter, and then processing the I and Q signals in a cumulant recovery processing system to obtain phase correction of the I and Q signals prior to demodulation. The final step is demodulating the two-dimensional communication signal. Phase correction is effected without regard to modulation type and independently of the demodulating step.

CROSS-REFERENCES TO RELATED APPLICATIONS

This is a division of application Ser. No. 08/755,775, filed Nov. 22,1996, now U.S. Pat. No. 6,018,317 entitled “Cochannel Signal ProcessingSystem,” which was a continuation-in-part of application Ser. No.08/459,902, filed Jun. 2, 1995, now abandoned having the same title, andof application Ser. No. 08/459,074, filed Jun. 2, 1995, now abandoned,entitled “Complex Phase Equalizer Using Cumulant Recovery Processing.”

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to signal processing systems and, moreparticularly, to apparatus and methods for receiving and processingsignals that share a common receiver frequency band at the same time,referred to as cochannel signals. Even two signals transmitted onslightly separated frequency bands may be “cochannel” signals as seen bya receiver operating to receive signals on a bandwidth that overlapsboth of the signals. In a variety of signal processing applications,there is a need to recover information contained in such multiple,simultaneously received signals. In the context of this invention, theword “recover” or “recovery” encompasses separation of the receivedsignals, “copying” the signals (i.e., retrieving any informationcontained in them), and, in some applications, combining signalsreceived over multiple paths from a single source. The “signals” may beelectromagnetic signals transmitted in the atmosphere or in space,acoustic signals transmitted through liquids or solids, or other typesof signals characterized by a time-varying parameter, such as theamplitude of a wave. In accordance with another aspect of the invention,signal processing includes transmission of cochannel signals.

In the environment of the present invention, signals are received by“sensors.” A sensor is an appropriately selected transducer forconverting energy contained in the signal to a more easily manipulatedform, such as electrical energy. In a radio communications application,electromagnetic signals are received by antennas and converted toelectrical signals for further processing. After separation of thesignals, they may be forwarded separately to transducers of a differenttype, such as loudspeakers, for converting the separated electricalsignals into audio signals. In some applications, the signal content maybe of less importance than the directions from which the signals werereceived, and in other applications the received signals may not beamenable to conversion to audible form. Instead, each recovered signalmay contain information in digital form, or may contain information thatis best understood by displaying it on a chart or electronic displaydevice. Regardless of the environment in which the present invention isemployed, it is characterized by multiple signals received by sensorssimultaneously at the same or overlapping frequencies, the need toseparate, recover, identify or combine the signals and, optionally, sometype of output transducer to put the recovered information in a moreeasily discernible form.

2. Description of Related Art

Separation and recovery of signals of different frequencies is a routinematter and is handled by appropriate filtering of the received signals.It is common knowledge that television and radio signals are transmittedon different frequency bands and that one may select a desired signal bytuning a receiver to a specific channel. Separation and recovery ofmultiple signals transmitted at different frequencies and receivedsimultaneously may be effected by similar means, using multiple tunedreceivers in parallel. A more difficult problem, and the one with whichthe present invention is concerned, is how to separate and copy signalsfrom multiple sources when the transmitted signals are at the same oroverlapping frequencies. A single sensor, such as an antenna, is unableto distinguish between two or more received signals at the samefrequency. However, antenna array technology provides for the separationof signals received from different directions. Basically, and as is wellunderstood by antenna designers, an antenna array can be electronically“steered” to transmit or receive signals to or from a desired direction.Moreover, the characteristics of the antenna array can be selectivelymodified to present “nulls” in the directions of signals other than thatof the signal of interest. A further development in the processing ofarray signals was the addition of a control system to steer the arraytoward a signal of interest. This feature is called adaptive arrayprocessing and has been known for at least two to three decades. See,for example, a paper by B. Widrow, P. E. Mantey, L. J. Griffiths and B.B. Goode, “Adaptive Antenna Systems,” Proceedings of the IEEE, vol. 55,no. 12, pp. 2143-2159, December 1967. The steering characteristics ofthe antenna can be rapidly switched to receive signals from multipledirections in a “time-sliced” manner. At one instant the antenna arrayis receiving a signal from one source and at the next instant, from adifferent source in a different direction, but information from themultiple sources is sampled rapidly enough to provide a complete recordof all the received signals. It will be understood that, althoughsteered antenna array technology was developed principally in thecommunications and radar fields, it is also applicable to the separationof acoustic and other types of signals.

In the communications field, signals take a variety of forms. Statedmost generally, a communication signal typically includes a carriersignal at a selected frequency, on which is impressed or modulated aninformation signal. There are a large number of different modulationschemes, including amplitude modulation, in which the amplitude of thesignal is varied in accordance with the value of an information signal,while the frequency stays constant, and frequency or phase modulation,in which the amplitude of the signal stays constant while its frequencyor phase is varied to encode the information signal onto the carrier.Various forms of frequency and phase modulation are often referred to asconstant modulus modulation methods, because the amplitude or modulus ofthe signal remains constant, at least in theory. In practice, themodulus is subject to distortion during transmission, and variousdevices, such as adaptive equalizers, are used to restore theconstant-modulus characteristic of the signal at a receiver. Theconstant modulus algorithm was developed for this purpose and laterapplied to antenna arrays in a process called adaptive beam forming Thefollowing references are provided by way for further background on theconstant modulus algorithm:

B. Agee, “The least-squares CMA: a new technique for rapid correction ofconstant modulus signals,” Proc. ICASSP-86, pp. 953-956, Tokyo, Japan,April 1986.

R. Gooch, and J. Lundell, “The CM array, an adaptive beamformer forconstant modulus signals,” Proc. ICASSP-86, pp. 2523-2526, Tokyo, Japan,April 1986.

J. Lundell, and B. Widrow, “Applications of the constant modulusadaptive algorithm to constant and non-constant modulus signals,” Proc.Twenty-Second Asilomar Conference on Signals, Systems, and Computers,pp. 432-436, Pacific Grove, Calif., November 1988.

B. G. Agee, “Blind separation and capture of communication signals usinga multi-target constant modulus beamformer,” Proc. 1989 IEEE MilitaryCommunications Conference, pp. 340-346, Boston, Mass., October 1989.

R. D. Hughes, E. H. Lawrence, and L. P. Withers, Jr., “A robust adaptivearray for multiple narrowband sources,” Proc. Twenty-Sixth AsilomarConference on Signals, Systems, and Computers, pp. 35-39, Pacific Grove,Calif., November 1992.

J. J. Shynk and R. P. Gooch, “Convergence properties of the multistageCMA adaptive beamformer,” Proc. Twenty-Seventh Asilomar Conference onSignals, Systems, and Computers, pp. 622-626, Pacific Grove, Calif.,November 1993.

The constant modulus algorithm works satisfactorily only for constantmodulus signals, such as frequency-modulated (FM) signals or variousforms of phase-shift keying (PSK) in which the phase is discretely orcontinuously varied to represent an information signal, but not foramplitude-modulated (AM) signals or modulation schemes that employ acombination of amplitude and phase modulation. There is a significantclass of modulation schemes used known as M-ary quadrature amplitudemodulation (QAM), used for transmitting digital data, whereby theinstantaneous phase and amplitude of the carrier signal represents aselected data state. For example, 16-ary QAM has sixteen distinctphase-amplitude combinations. The “signal constellation” diagram forsuch a scheme has sixteen points arranged in a square matrix and lyingon three separate constant-modulus circles. A signal constellationdiagram is a convenient way of depicting all the possible signal statesof a digitally modulated signal. In such a diagram, phase is representedby angular position and modulus is represented by distance from anorigin.

The constant modulus algorithm has been applied with limited success toa 16-ary QAM scheme, because it can be represented as three separateconstant-modulus signal constellations. However, for higher orders ofQAM the constant modulus algorithm provides rapidly decreasing accuracy.For suppressed-carrier AM, the constant modulus approach failscompletely in trying to recover cochannel AM signals. If there aremultiple signals, the constant modulus algorithm yields signals with“cross-talk,” i.e. with information in the two signals being confused.For a single AM signal in the presence of noise, the constant-modulusalgorithm yields a relatively noisy signal.

Because antenna arrays can be steered electronically to determine thedirections of signal sources, it was perhaps not surprising that onewell known form signal separator available prior to the presentinvention used direction finding as its basis. The approach is referredto as DF-aided copy, where DF means direction finding. This is anopen-loop technique in which steering vectors that correspond toestimated signal source bearings are first determined; then used toextract waveforms of received signals. However, the direction findingphase of this approach requires a knowledge of the geometry andperformance characteristics of the antenna array. Then steering vectorsare fed forward to a beamformer, which nulls out the unwanted signalsand steers one or more antenna beam(s) toward each selected source.

Prior to the present invention, some systems for cochannel signalseparation used direction-finding (DF)-beamforming. Such systemsseparate cochannel signals by means of a multi-source (or cochannel)super-resolution direction finding algorithm that determines steeringvectors and directions of arrival (DOAs) of multiple simultaneouslydetected cochannel signal sources. An algorithm determines beam-formingweight vectors from the set of steering vectors of the detected signals.The beamforming weight vectors are then used to recover the signals. Anyof several well-known multi-source super-resolution DF algorithms can beused in such a system. Some of the better known ones are usuallyreferred to by the acronyms MUSIC (MUltiple SIgnal Classification),ESPRIT Estimation of Signal Parameters via Rotational InvarianceTechniques), Weighted Subspace Fitting (WSF), and Method of DirectionEstimation (MODE).

MUSIC was developed in 1979 simultaneously by Ralph Schmidt in theUnited States and by Georges Bienvenu and Lawrence Kopp in France. TheSchmidt work is described in R. O. Schmidt, “Multiple emitter locationand signal parameter estimation,” Proc. RADC Spectrum EstimationWorkshop, pp. 243-258, Rome Air Development Center, Griffiss Air ForceBase, NY, October 3-5, 1979. The Bienvenu work is described in G.Bienvenu and L. Kopp, “Principe de la goniometrie passive adaptative,”Proc. Collogue GRETSI, pp. 106/1-106/10, Nice, France, May 1979. MUSIChas been extensively studied and is the standard against which othersuper-resolution DF algorithms are compared.

ESPRIT is described in many publications in the engineering signalprocessing literature and is the subject of U.S. Pat. No. 4,750,147entitled “Method for estimating signal source locations and signalparameters using an array of sensor pairs,” issued to R. H. Roy III etal. ESPRIT was developed by Richard Roy, III, Arogyaswami Paulraj, andProf. Thomas Kailath at Stanford University. It was presented as asuper-resolution algorithm for direction finding in the following seriesof publications starting in 1986:

A. Paulraj, R. Roy, and T. Kailath, “A subspace rotation approach tosignal parameter estimation,” Proc. IEEE, vol. 74, no. 4, pp. 1044-1045,July 1986.

R. Roy, A. Paulraj, and T. Kailath, “ESPRIT—A subspace rotation approachto estimation of parameters of cisoids in noise,” IEEE Trans. Acoust.,Speech, and Signal Processing, vol. ASSP-34, no. 5, pp. 1340-1342,October 1986.

R. H. Roy, ESPRIT—Estimation of Signal Parameters via RotationalInvariance Techniques, doctoral dissertation, Stanford University,Stanford, Calif., 1987.

R. Roy and T. Kailath, “ESPRIT—Estimation of signal parameters viarotational invariance techniques,” IEEE Trans. Acoust., Speech, andSignal Processing, vol. ASSP-37, no. 7, pp. 984-995, July 1989.

B. Ottersten, R. Roy, and T. Kailath, “Signal waveform estimation insensor array processing,” Proc. Twenty-Third Asilomar Conference onSignals, Systems, and Computers, pp. 787-791, Pacific Grove, Calif.,November 1989.

R. Roy and T. Kailath, “ESPRIT—Estimation of signal parameters viarotational invariance techniques,” Optical Engineering, vol. 29, no. 4,pp. 296-313, April 1990.

MUSIC and ESPRIT both require the same “narrowband array assumption,”which is further discussed below in the detailed description of theinvention, and both are modulation independent, a feature shared by allcochannel signal separation and recovery techniques that are based onthe DF-beamforming method.

ESPRIT calculates two N-by-N covariance matrices, where N is the numberof antenna elements, and solves a generalized eigenvalue problemnumerically (instead of using a calibration table search, as MUSICdoes). It does this for every block of input samples. MUSIC calculates asingle N-by-N covariance matrix, performs an eigendecomposition, andsearches a calibration table on every block of input array samples(snapshots).

MUSIC and ESPRIT have a number of shortcomings, some of which arediscussed in the following paragraphs.

ESPRIT was successfully marketed based on a single, key advantage overMUSIC. Unlike MUSIC, ESPRIT did not require array calibration. InESPRIT, the array calibration requirement was eliminated, and adifferent requirement on the antenna array was substituted. The newrequirement was that the array must have a certain geometrical property.Specifically the array must consist of two identical sub-arrays, one ofwhich is offset from the other by a known displacement vector. Inaddition, ESPRIT makes the assumption that the phases of receivedsignals at one sub-array are related to the phases at the othersub-array in an ideal theoretical way.

Another significant disadvantage of ESPRIT is that, although it purportsnot to use array calibration, it has an array manifold assumption hiddenin the theoretical phase relation between sub-arrays. “Array manifold”is a term used in antenna design to refer to a multiplicity of physicalantenna parameters that, broadly speaking, define the performancecharacteristics of the array.

A well known difficulty with communication systems, especially in anurban environment, is that signals from a single source may be receivedover multiple paths that include reflections from buildings and otherobjects. The multiple paths may interpose different time delays, phasechanges and amplitude changes on the transmitted signals, renderingreception more difficult, and transmission uncertain. This difficulty isreferred to as the multipath problem. It is one that has not beenadequately addressed by signal processing systems of the prior art.

Neither MUSIC nor ESPRIT can operate in a coherent multipath environmentwithout major added complexity. A related problem is that, in a signalenvironment devoid of coherent multipath, no DF-beamforming method canseparate signals from sources that are collinear with the receivingarray, i.e. signal sources that are in line with the array and have zeroangular separation. Even in a coherent multipath environment,DF-beamforming methods like MUSIC and ESPRIT cannot separate and recovercochannel signals from collinear sources.

Another difficulty with ESPRIT is that it requires two antennasub-arrays and is highly sensitive to mechanical positioning of the twosub-arrays, and to the electromagnetic matching of each antenna in onesub-array with its counterpart in the other sub-array. Also ESPRITrequires a 2N-channel receiver, where N is the number of antennaelements, and is highly sensitive to channel matching.

Another significant drawback in both MUSIC and ESPRIT is that they failabruptly when the number of signals detected exceeds the capacity, N,equal to the number of antennas in the case of MUSIC, or half the numberof antennas in the case of ESPRIT.

A fundamental problem with both MUSIC and ESPRIT is that they useopen-loop feed-forward computations, in which errors in the determinedsteering vectors are uncorrected, uncorrectable, and propagate intosubsequent calculations. As a consequence of the resultant inaccuratesteering vectors, MUSIC and ESPRIT have poorcross-talk rejection, asmeasured by signal-to-interference-plus-noise ratio (SINR) at the signalrecovery output ports.

ESPRIT is best suited to ground based systems where its antennarequirements are best met and significant computational resources areavailable. MUSIC has simpler antenna array requirements and lends itselfto a wider range of platforms, but also needs significant computationalresources.

Another limitation of most signal recovery systems of the prior art isthat they rely on first-order and second-order statistical moments ofthe received signal data. A moment is simply a statistical quantityderived from the original data by mathematical processing at some level.An average or mean value of the several signals received at a given timeis an example of a first-order moment. The average of the squares of thesignal values (proportional to signal powers) is an example of asecond-order moment. Even if one considers just one signal and a noisecomponent, computing the average of the sum of the squares produces across-term involving the product of signal and noise components.Typically, engineers have managed to find a way to ignore the cross-termby assuming that the signal and the noise components are statisticallyindependent. At a third-order level of statistics, one has to assumethat the signal and noise components have zero mean values in order toeliminate the cross-terms in the third-order moment. For thefourth-order and above, the computations become very complex and are noteasily simplified by assumptions. In most prior art signal analysissystems, engineers have made the gross assumption that the nature of allsignals is Gaussian and that there is no useful information in thehigher-order moments. Higher-order statistics have been long recognizedin other fields and there is recent literature suggesting theirusefulness in signal recovery. Prior to this invention, cumulant-basedsolutions have been proposed to address the “blind” signal separationproblem, i.e. the challenge to recover cochannel signals withoutknowledge of antenna array geometry or calibration data. See, forexample, the following references:

J.-F. Cardoso, “Source separation using higher order moments,” Proc.ICASSP-89, pp. 2109-2112, Glasgow, Scotland, May 1989.

J.-F. Cardoso, “Eigen-structure of the fourth-order cumulant tensor withapplication to the blind source separation problem,” Proc. ICASSP-90,pp. 2655-2658, Albuquerque, N. Mex., April 1990.

J.-F. Cardoso, “Super-symmetric decomposition of the fourth-ordercumulant tensor: blind identification of more sources than sensors,”Proc. ICASSP-91, pp. 3109-3112, Toronto, Canada, May 1991.

J.-F. Cardoso, “Higher-order narrowband array processing,” InternationalSignal Processing Workshop on Higher Order Statistics, pp. 121-130,Chamrousse-France, July 10-12, 1991.

J.-F. Cardoso, “Blind beamforming for non-Gaussian sources,” IEEProceedings Part F, vol. 140, no. 6, pp. 362-370, December 1993.

P. Comon, “Separation of stochastic processes,” Proc. Vail Workshop onHigher-Order Spectral Analysis, pp. 174-179, Vail, Colo., USA, June1989.

P. Comon, “Independent component analysis,” Proc. of Intl. Workshop onHigher-Order Statistics, pp. 111-120, Chamrousse, France, 1991.

P. Comon, C. Jutten, and J. Herault, “Blind separation of sources, partII: problems statement,” Signal Processing, vol. 24, no. 1, pp. 11-20,July 1991.

E. Chaumette, P. Comon, and D. Muller, “ICA-based technique forradiating sources estimation: application to airport surveillance,” IEEProceedings Part F, vol. 140, no. 6, pp. 395-401, December 1993.

Z. Ding, “A new algorithm for automatic beamforming,” Proc. Twenty-FifthAsilomar Conference on Signals, Systems, and Computers, pp. 689-693,Pacific Grove, Calif., November 1991.

M. Gaeta and J.-L. Lacoume, “Source separation without a-prioriknowledge: the maximum likelihood solution,” Proc. EUSIPCO, pp. 621-624,1990.

E. Moreau, and O. Macchi, “New self-adaptive algorithms for sourceseparation based on contrast functions,” Proc. IEEE SP Workshop onHigher-Order Statistics, pp. 215-219, Lake Tahoe, USA, June 1993.

P. Ruiz, and J. L. Lacoume, “Extraction of independent sources fromcorrelated inputs: a solution based on cumulants,” Proc. Vail Workshopon Higher-Order Spectral Analysis, pp. 146-151, Vail, Colo., USA, June1989.

E. H. Satorius, J. J. Mulligan, Norman E. Lay, “New criteria for blindadaptive arrays,” Proc. Twenty-Seventh Asilomar Conference on Signals,Systems, and Computers, pp. 633-637, Pacific Grove, Calif., November1993.

L. Tong, R. Liu, V. Soon, and Y. Huang, “Indeterminacy andidentifiability of blind identification,” IEEE Trans. Circuits andSystems, vol. 38, pp. 499-509, May 1991.

L. Tong, Y. Inouye and R. Liu, “Waveform preserving blind estimation ofmultiple independent sources,” IEEE Trans. Signal Processing, vol. 41,no. 7, pp. 2461-2470, July 1993.

However, all of these approaches to blind signal recovery address thestatic case in which a batch of data is given to a processor, which thendetermines the steering vectors and exact waveforms. These priorapproaches do not have the ability to identify new sources that appearor existing sources that are turned off. In addition, previouslyproposed algorithms require multiple levels of eigendecomposition ofarray covariance and cumulant matrices. Their convergence to reliablesolutions depends on the initialization and utilization of the cumulantmatrices that can be derived from array measurements. Furthermore,previous cumulant-based algorithms have convergence problems in the caseof identically modulated sources in general.

Ideally, a system for receiving and processing multiple cochannelsignals should make use of statistics of the measurements, and shouldnot need to rely on knowledge of the geometry or array manifold of thesensors, i.e., the array calibration data. Also, the system should beable to receive and process cochannel signals regardless of theirmodulation or signal type, e.g. it should not be limited toconstant-modulus signals. More generally, the ideal cochannel signalprocessing system should not be limited to any modulation properties,such as baud rate or exact center frequency. Any system that is limitedby these properties has only a limited range of source types that can beseparated, and is more suitable for interference suppression insituations where the desired signal properties are well known. Anotherdesirable property of the ideal cochannel signal receiving andprocessing system is that it should operate in a dynamic way,identifying new signal sources that appear and identifying sources thatdisappear. Another desirable characteristic is a very high speed ofoperation allowing received signals to be processed in real time. Aswill shortly become apparent, the present invention meets and exceedsthese ideal characteristics for cochannel signal processing.

SUMMARY OF THE INVENTION

The present invention resides in a system or method for processingcochannel signals received at a sensor array and producing desiredrecovered signals or parameters as outputs. In the context of thisspecification, “cochannel” signals are that overlap in frequency, asviewed from a receiver of the signals. Even signals that are transmittedin separate, but closely spaced, frequency bands may be cochannelsignals as viewed from a receiver operating in bandwidth wide enough tooverlap both of the signals. A key aspect of the invention is that it iscapable of separating and recovering multiple cochannel signals veryrapidly using only sensor array signals, without knowledge of sensorarray geometry and array manifold, (e.g. array calibration data), andwithout regard to the signal type or modulation. If array calibrationdata are available, the system also provides direction-of-arrivalparameters for each signal source. The invention inherently combinescoherent multipath components of a received signal and as a resultachieves improved performance in the presence of multipath. Oneembodiment of the invention also includes a transmitter, which makes useof estimated generalized steering vectors generated while separating andrecovering received signals, in order to generate appropriate steeringvectors for transmitted signals, to ensure that transmitted signalsintended for a particular signal source traverse generally the same pathor paths that were followed by signals received from the same signalsource.

Briefly, and in general terms, the system of the invention comprises asignal receiving system, including means for generating a set ofconditioned receiver signals from received signals of any modulation ortype; an estimated generalized steering vector (EGSV) generator, forcomputing an EGSV that results in optimization of a utility functionthat depends on fourth or higher even-order statistical cumulantsderived from the received signals, the EGSV being indicative of acombination of signals received at the sensors from a signal source; anda supplemental computation module, for deriving at least one outputquantity of interest from the conditioned receiver signals and the EGSV.

The basic invention as described in the preceding paragraph employs oneof three basic methods for computing EGSVs: two iterative methods andone direct computation method. In the first iterative method, the systemincludes a linear combiner, for repeatedly computing a single channelcombined signal from the conditioned receiver signals and an EGSV; meansfor supplying an initial EGSV to the linear combiner, to produce theinitial output of a single channel combined signal; an EGSV computationmodule, for computing successive values of the EGSV from successivevalues of the single channel combined signal received from the linearcombiner and the conditioned receiver signals; and means for feeding thesuccessive values of the EGSV back to the linear combiner for successiveiteration cycles. Also included is means for terminating iterativeoperation upon convergence of the EGSV to a sufficiently accurate value.

If the second iterative method is used, the system includes across-cumulant matrix computation module, for generating a matrix ofcross-cumulants of all combinations of the conditioned receiver signals;a structured quadratic form computation module, for computing successivecumulant strength functions derived from successive EGSVs and thecross-cumulant matrix; means for supplying an initial EGSV to thestructured quadratic form computation module, to produce the initialoutput of a cumulant strength function; an ESGV computation module, forgenerating successive EGSVs from successive cumulant strength functionsreceived from the structured quadratic form computation module; meansfor feeding the successive values of the EGSV back to the structuredquadratic form computation module for successive iteration cycles; andmeans for terminating iterative operation upon convergence of the EGSVto a sufficiently accurate value.

Finally, if the direct computation method is used, the system includes across-cumulant matrix computation module, for generating a matrix ofcross-cumulants of all combinations of the conditioned receiver signals;and an EGSV computation module for computing the EGSV directly from thecross-cumulant matrix by solving a fourth degree polynomial equation.

Regardless which of the foregoing variants is employed, signalprocessing may employ one of several different cumulant recovery (CURE)techniques. In a first of these techniques, the means for generating theset of conditioned signals includes a covariance matrix computationmodule, an eigendecomposition module for generating the eigenstructureof the covariance matrix and an estimate of the number of signalsources, and a transformation matrix for conditioning the receiversignals. An EGSV generator then employs signals output by theeigendecomposition module to compute EGSVs. This technique is referredto in this specification as the eigenCURE or eCURE system.

An alternate processing technique uses covariance inversion of thereceived signals and is referred to as the CiCURE system. In thisapproach, the means for generating the set of conditioned signalsincludes a covariance matrix computation module and a matrixdecomposition module, for generating the inverse covariance matrix and atransformation matrix for conditioning the receiver signals. An EGSVgenerator then employs signals output by the eigendecomposition moduleto compute EGSVs. The system further includes a beamformer, forgenerating a recovered signal from the receiver signals by using theEGSV(s) and the matrix obtained from the matrix decomposition module.

Yet another processing technique is referred to as pipelined cumulantrecovery, or pipeCURE. The means for generating the set of conditionedsignals includes a covariance matrix computation module, aneigendecomposition module for generating an estimate of the number ofsignal sources, a transformation matrix for conditioning the receiversignals, and an eigenstructure derived from the receiver signals. Again,the EGSV generator employs signals output by the eigendecompositionmodule to compute EGSVs. Processing is on a block-by-block basis, andthe system further comprises a multiple port signal recovery unit,including means for matching current EGSVs with EGSVs from a prior datablock to impose waveform continuity from block to block.

Another variant that can be used in any of these processing techniquesinvolves the manner in which initial EGSVs are computed at the start ofprocessing a new block of data. In accordance with this aspect of theinvention, the initial values of EGSVs for each new processing block arecomputed by combining a prior block EGSV and a cumulant vector derivedfrom the utility function used in the EGSV generator. More specifically,the means for combining takes the sum of the prior block EGSV multipliedby a first factor, and the cumulant vector multiplied by a secondfactor. The first and second factors may be selected to provide aninitial EGSV that anticipates and compensates for movement of a signalsource.

In a practical embodiment of the invention, the system functions toseparate a plurality (I of received cochannel signals. If the firstiterative method is employed, there are multiple EGSV generators (P innumber), including P EGSV computation modules and P linear combiners,for generating an equal plurality (P) of EGSVs associated with P signalsources. The supplemental computation module functions to recover Pseparate received signals from the P generalized steering vectors andthe conditioned receiver signals. More specifically, the supplementalcomputation module includes a recovery beamformer weight vectorcomputation module, for generating from all of the EGSVs a plurality (P)of receive weight vectors, and a plurality (P) of recovery beamformers,each coupled to receive one of the P receive weight vectors and theconditioned receiver signals, for generating a plurality (P) ofrecovered signals.

For recovery of multiple signals using the second iterative method,there is a plurality (P) of EGSV generators, including P EGSVcomputation modules and P structured quadratic form computation modules,for generating an equal plurality (P) of EGSVs associated with P signalsources. Again, the supplemental computation module includes a recoverybeamformer weight vector computation module, for generating from all ofthe EGSVs a plurality (P) of receive weight vectors, and a plurality (P)of recovery beamformers, each coupled to receive one of the P receiveweight vectors and the conditioned receiver signals, for generating aplurality (P) of recovered signals.

If the direct processing method is used to separate two signals, theESGV computation module generates two EGSVs from the cross-cumulantmatrix data; and the supplemental computation module functions torecover two separate received signals from the two generalized steeringvectors and the conditioned receiver signals. The supplementalcomputation module includes a recovery beamformer weight vectorcomputation module, for generating from both of the EGSVs two receiveweight vectors, and two recovery beamformers, each coupled to receiveone of the receive weight vectors and the conditioned receiver signals,for generating two recovered signals.

Although the system of the invention operates in a “blind” sense,without knowledge of the geometry or calibration data of the sensorarray, it will also function as a direction finder if array calibrationdata are available. Hence, in one embodiment of the invention, thesystem functions to derive the direction of arrival (DOA) of a receivedsignal; and the supplemental computation module includes a memory forstoring sensor array calibration data, and means for deriving the DOA ofa received signal from its associated steering vector and the storedsensor array calibration data. More specifically, the sensor arraycalibration data includes a table associating multiple DOA values withcorresponding steering vectors; and the means for deriving the DOAincludes means for performing a reverse table lookup function to obtainan approximated DOA value from a steering vector supplied by thegeneralized steering vector generator. The means for deriving the DOAmay also include means for interpolating between two DOA values toobtain a more accurate result.

In another important embodiment of the invention, the supplementalcomputation module of the signal processing system also includes atransmitter, for generating transmit signal beamformer weights from thereceived signal beamformer weights, and for transmitting signalscontaining information in a direction determined by the transmit signalbeamformer weights.

Other aspects of the invention pertain to various application of thebasic cumulant recovery (CURE) signal processing engine described above.Some of these applications are summarized in the following paragraphs.

An important application of the invention is in two-way radiocommunication. Because CURE processing generates an estimatedgeneralized steering vector not necessarily for each received signal,but for each signal source, the invention provides an important benefitwhen used in multipath conditions. Signals reaching a receiving antennaarray over multiple paths will be combined in the CURE system if thereceived components are coherent, and the resultant generalized steeringvector represents the combined effect of all the coherent multipathsignals received at the antenna. This feature has a number ofadvantages. First, a radio receiving system using the CURE system isinherently immune to multipath problems encountered by conventionalreceivers. Second, by using generalized steering vectors, there can bean associated generalized null in the antenna directivity pattern, whichcan be used to null out an interfering signal having multipath structurein favor of a cochannel signal from another source. Third, the signalrecovery method provides a diversity gain in the presence of multipathcomponents, such that a stronger combined signal is received as comparedwith a system that discards all but one component. Fourth, thegeneralized steering vector concept allows multiple cochannel signals tobe received and transmitted in the presence of multipath effects. Fifth,cochannel signal sources that are collinear with the receiver sensorarray can be received and separated if there are multipath components.

In another aspect of the invention, the CURE signal separation systemcan be used to separate signals transmitted in different modes over a“waveguide,” by which is meant any bounded propagation medium, such as amicrowave waveguide, an optical waveguide, a coaxial cable, or even atwisted pair of conductors. Although the transmission modes may becomemixed in the waveguide, the original signals are easily recovered in theCURE system.

In still another aspect of the invention, the CURE signal separationsystem can be used to separate signals recorded on closely space trackson a recording medium. Crosstalk between the signals on adjacent tracksis eliminated by using the CURE system to effect recovery.

In yet another aspect of the invention, the CURE signal separationsystem can be used to extend the effective dynamic range of a receiversystem.

In a further aspect of the invention, the CURE signal separation systemcan be used to perform a complex phase equalization functionautomatically, without knowledge of the amount of phase correction thatis needed.

The CURE system may be modified to compensate for moving signal sources,and may also be modified to handle a wideband signal separation problem.The wideband signal separator includes multiple narrowband CURE systems,means for decomposing a wide band of signals into multiple narrowbandsfor processing, and means for combining the narrow bands again.

Other aspects and advantages of the invention will become apparent fromthe following more detailed description, taken together with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a taxonomy diagram depicting the interrelationships of theinvention's various cochannel signal processing methods.

FIG. 2A is a block diagram of the cumulant recovery (CURE) system of theinvention in general form, depicting a first iterative method.

FIG. 2B is a block diagram similar to FIG. 2A, but depicting a seconditerative method.

FIG. 2C is a block diagram similar to FIG. 2A, depicting a directcomputation method.

FIG. 3A is a block diagram similar to FIG. 2A, but modified to depicthow the invention functions to separate multiple received cochannelsignals.

FIG. 3B is a block diagram similar to FIG. 3A, but depicting the firstiterative method.

FIG. 3C is a block diagram similar to FIG. 3A, but depicting the directcomputation method.

FIG. 4A, is a block diagram of a supplemental computation module forsignal recovery, shown in FIGS. 3A, 3B and 3C.

FIG. 4B is a block diagram of a supplemental computation module for usein conjunction with the system of FIGS. 2A, 2B or 2C, to provide adirection finding function.

FIG. 5 is a block diagram similar to FIG. 2A, but modified to depict areceiver/transmitter function.

FIG. 6 is a block diagram similar to FIG. 3A, but modified to show inputof signals from a signal “waveguide.”

FIG. 7 is a block diagram showing in more detail the functions performedin accordance with the two iterative methods for cochannel signalseparation and recovery.

FIG. 8A is simplified block diagram of a cochannel signal recoverysystem in accordance with the present invention.

FIG. 8B is a hardware block diagram similar to FIG. 8A, of one preferredembodiment of the invention.

FIG. 9 (comprising FIGS. 9A and 9B) is another block diagram of thesystem of FIG. 7, with some of the subsystem functions recited in eachof the blocks.

FIG. 10 is a block diagram of the preprocessor computer of FIG. 7.

FIG. 11 (comprising FIGS. 11A and 11B) is a block diagram of a signalextraction port in an active state.

FIG. 12 is a block diagram of the multiple port recovery unit of FIG. 9,including multiple signal extraction ports and an orthogonalizer.

FIG. 13 (comprising FIGS. 13A, 13B and 13C) is a block diagram of thesignal recovery controller of FIG. 7.

FIG. 14 is a block diagram of a signal extraction port in the inactivestate.

FIG. 15 is a block diagram of signal recovery system using covarianceinversion cumulant recovery (CiCURE).

FIG. 16 is a block diagram of a signal extraction port in the activestate using the CiCURE system illustrated in FIG. 15.

FIG. 17 is a block diagram of an alternate embodiment of the inventionreferred to as pipeCURE.

FIG. 18 is a schematic diagram depicting a processing difficultyassociated with moving signal sources.

FIGS. 19 and 20 are vector diagrams depicting the effect of usingα-βCURE and μCURE updating to initialize a block of samples for eCUREprocessing in the moving source situation such as shown in FIG. 18.

FIG. 21 is a schematic diagram showing an overload condition in whichthere are more signal sources that antenna elements.

FIG. 22 is a diagram of an antenna array directivity pattern forhandling the situation shown in FIG. 21.

FIG. 23 is a schematic diagram depicting multipath propagation pathsfrom a transmitter to a receiver array.

FIG. 24 is a schematic diagram depicting how the system of the inventionhandles coherent multipath signals in the presence of interference.

FIG. 25 is a schematic diagram depicting how the system of the inventionhandles non-coherent multipath signals in the presence of interference.

FIG. 26 is a schematic diagram depicting how the system of the inventionhandles receiving a desired signal in the presence of an interferencesignal with multipath components and a second interference source.

FIG. 27 is a schematic diagram depicting the invention as used in therecovery of a received signal in the presence of a strong localtransmitter.

FIG. 28 is a schematic diagram depicting the invention as used in therecovery of a weak received signal in the presence of a strong jammingsignal located nearby.

FIG. 29 is a schematic diagram depicting portions of a cellulartelephone communication system of the prior art.

FIG. 30 is a schematic diagram depicting how a communication system ofthe prior art separates cochannel signals.

FIG. 31 is a schematic diagram depicting how the present inventionoperates in the environment of a cellular telephone communicationsystem.

FIG. 32 is a block diagram showing major portions of a transmitter asused in conjunction with the system of the present invention.

FIG. 33 is a block diagram showing a transmitter similar to the one inFIG. 32, but with more detail of some aspects of the apparatus.

FIGS. 34A and 34B are diagrams depicting operation of the invention inrecovery of dual-polarized signals.

FIG. 35 is a schematic diagram depicting the invention as used inconjunction with an optical fiber network.

FIG. 36 is a schematic diagram depicting the invention as used incopy-aided direction finding.

FIG. 37 is a schematic diagram depicting the invention as used inextending the dynamic range of a receiver.

FIG. 38 is a schematic diagram depicting the invention as used in diskrecording system.

FIG. 39 is a schematic diagram depicting the invention as used to effectautomatic phase rotation equalization of a QAM signal.

FIG. 40A is a signal constellation diagram for a QAM system with fouramplitude-phase states by way of illustration.

FIG. 40B is a diagram similar to FIG. 40A, showing a phase rotation of θas a result of propagation of the signal to the receiver.

FIG. 41 is a block diagram showing extension of the cumulant recoverysystem of the invention to cover a wide frequency band.

FIG. 42 is a set of graphs demonstrating how the probability densityfunction (PDF) of the sum of three random variables approaches Gaussianform.

FIG. 43 is a block diagram showing the function of a cumulant strengthbased processor in separating one of many cochannel sources.

FIG. 44 is a graph showing regions of convergence as a function ofinitialization and source statistics, and demonstrating an unstablenonconvergence condition.

FIGS. 45 and 46 are spectra of first and second frequency-modulated (FM)sources as used in simulation experiments using the invention.

FIG. 47 is a spectrum of an amplitude-modulated (AM) source as used inthe simulation experiments.

FIG. 48 is a graph of direction estimates obtained from active ports inthe simulation experiments.

FIG. 49 is a graph of samples of an original speech waveform as used inthe simulation experiments.

FIG. 50 is a graph of samples of a recovered waveform as used in thesimulation experiments.

FIG. 51 is a graph showing magnitudes of a port output signal, as usedin the simulation experiments.

FIG. 52 is a spectrum of measurements at a first sensor in thesimulation experiments.

FIG. 53 is a spectrum of a port output signal, as used in the simulationexperiments.

FIG. 54 is a graph of results from a number-of-sources estimator, asused in the simulation experiments.

FIG. 55 is a graph of bearing estimates from the optional copy-aided DFunit, which obtains the steering vectors from active ports, as used inthe simulation experiments.

FIGS. 56A and 56B shows a pair of graphs comparing an original speechwaveform with its estimate obtained from an active port in thesimulation experiments.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

1.0 Introduction:

Because the present invention encompasses a number of different butrelated concepts and applications, and because the key signal processingconcepts of the invention can be implemented in several differentembodiments, this detailed description is divided into sections andsubsections, each of which covers a different inventive concept,specific embodiment, or practical application of the invention. Thefollowing is a table of contents of this description of the preferredembodiments:

1.0 Introduction

2.0 Overview of the Concept of the Invention

2.1 Signal Separation Concept

2.2 Signal Separation Concept in the Multipath Environment

2.3 Direction Finding Concept

2.5 Transmitter/Receiver Concept

2.6 Concept of Separation of Signals in a “Waveguide”

3.0 Preferred Embodiment Using EigenCumulant Recovery (eCURE) System

3.1 Overview and System Hardware

3.2 Preprocessing

3.3 Operation of an Active Signal Extraction Port

3.4 The Signal Recovery Controller

3.5 The Orthogonalizer

3.6 Operation at an Inactive Port

4.0 Alternate Embodiment Using Covariance Inversion Cumulant Recovery(CiCURE) System

5.0 Alternate Embodiment Using Pipelined Cumulant Recovery (pipeCURE)System

5.1 Overview of the pipeCURE Signal Separator

5.2 Preprocessor Unit

5.3 Cumulant Matrix Computer

5.4 Multiple Port Signal Recovery Unit

6.0 Steering Vector Tracking Method for Situations Having RelativeMotion

7.0 Alternate Embodiment Using Direct Computation

8.0 Separation Capacity and Performance When Overloaded

9.0 Performance of the Invention in the Presence of Multipath

9.1 Performance Against Coherent Multipath

9.2 Performance Against Noncoherent Multipath

9.3 Performance Against Mixtures of Coherent and Noncoherent Multipath

10.0 Signal Recovery in the Presence of Strong Interfering Signals

11.0 Diversity Path Multiple Access (DPMA) Communication

11.1 History and Prior Art of Multiple Access Communication

11.2 A New Method of Multiple Access Communication

12.0 Application to Two-Way Wireless Communication Systems

12.1 Transmit Beamforming

13.0 Application to Recovery of Multimode Signals

14.0 Application to Separation of Signals Transmitted Over “Waveguide”

15.0 Application to Radio Direction Finding

16.0 Application to Extending the Dynamic Range of Receiving Systems

17.0 Application to High Density Recording

18.0 Application to Complex Phase Equalization

19.0 Extension to Wideband Signal Separation

19.1 Partitioning Wideband Measurements to Narrowbands

19.2 Signal Separation in Narrowbands

19.3 Combining Narrowbands

20.0 Conclusion

21.0 Mathematical Basis for the Invention

2.0 Overview of the Concept of the Invention:

The present invention resides in a system and method for processingsignals received by a sensor receiving system having an array of sensorelements. The system is capable of receiving and processing a signalfrom a single source, and for purposes of explanation the system willsometimes be described as receiving and processing just one signal. Itwill, however, shortly become clear that the system is capable of, andbest suited for, receiving and processing signals from multiple sources,and that the multiple signals may utilize overlapping signal frequencybands as viewed at the receiver, i.e. they may be cochannel signals. Asdiscussed above in the “background of the invention” section, detectionand processing of cochannel signals presents difficulties that have notbeen adequately addressed by signal processing systems of the prior art.

One central concept that makes the various aspects of overall inventionpossible is the ability to determine accurate estimates of generalizedsteering vectors of the source signals incident on a sensor array or,more generally, estimates of the linear combinations that represent therelative amplitudes of the source signals in each channel of amultichannel signal stream. The meaning of the term “generalizedsteering vector” will become better understood as this descriptionproceeds. A definition for present purposes is that the generalizedsteering vector associated with a particular signal source representsthe weighted sum of ordinary steering vectors corresponding to allmultipath components of signals from that source. An ordinary steeringvector is the value of the array manifold at a single anglecorresponding to a source's DOA. Multipath components arise when asignal from a single source reaches a receiver over multiple propagationpaths.

The invention encompasses three distinct methods for determiningestimates of the generalized steering vectors of source signals. FIG. 1illustrates the taxonomy and relationships among the methods. Two of themethods for determining the estimated generalized steering vectors(EGSVs) involve iterative computations. The first of these, indicated byreference numeral 10, is called the “beamform—cross-cumulant” method. Inthis method, an initial EGSV is iteratively updated and improved by amethod that involves using the initial EGSV as a complex weight vectorin a linear beamformer that operates on sensor signals (i.e., signalmeasured at a sensor array, to be discussed shortly with reference toother drawing figures). Cross-cumulants are computed, specificallybetween the output signal from the beamformer used in this method andeach of the beamformer's input signals.

The term “cumulant” is defined more completely in later sections of thisspecification, but for purposes of this general discussion it issufficient to note that cumulants are fourth-order (or highereven-order) statistical moments of the received signals. Thecross-cumulants are formed into a vector which, upon normalization,becomes the next EGSV. This iteration cycle is repeated untilconvergence is attained, based on a predefined convergence test or on aselected number of iteration cycles. Upon convergence, the EGSV willhave converged to the generalized steering vector of one of the incidentsource signals.

The second iterative method for obtaining EGSVs is referred to as the“C- matrix” method and is indicated at 12 in FIG. 1. In this method, aparticular matrix of cross-cumulants among the sensor signals iscalculated first, before the iterative process starts. There are nosubsequent calculations of cumulants within the iterative cycle.Instead, an initial EGSV is used to calculate a structured quadraticform, which yields a quantity called “cumulant strength.” The EGSV isadjusted to maximize the cumulant strength by means of an iterativeoptimization procedure. Upon convergence, the EGSV will have convergedto the generalized steering vector of one of the incident sourcesignals.

The third method of finding the EGSVs is non-iterative. It is referredto as the “direct computation” or “analytic” method and is indicated at14 in FIG. 1. This method can only recoverup to two incident signals,but is unrestricted as to the number of sensor array elements orprocessing channels. The method first computes the same matrix ofcross-cumulants among the sensor signals as is used in the seconditerative method. However, a fourth degree polynomial equation is solvedinstead of optimizing a structured quadratic form. The solutions to thefourth degree polynomial yield the final EGSVs directly withoutiteration.

In addition to the three methods for generating the EGSVs of the sourcesignals, the present invention has three alternative basic algorithmicstructures on which the preferred embodiments of the invention arebased. FIG. 1 shows diagrammatically that the EGSVs generated by eitherof the three methods (10, 12 or 14) are coupled over line 16 to aselected one of three algorithmic structures 18, referred to as CiCURE,eCURE, and pipeCURE. As will be further explained below, the acronymCURE stands for cumulant recovery. The terms CiCURE, eCURE and pipeCURErefer to specific embodiments of the invention known as covarianceinversion CURE, eigenCURE and pipelined CURE, respectively. Eachstructure encompassed by the block 18 in FIG. 1 can employ any of thethree methods (10, 12 or 14) described above for determining the sourcesignal EGSVs.

By way of example, a subsequent section of this specification (Section3.0) describes a preferred embodiment consisting of the eCUREalgorithmic structure and the beamform—cross-cumulant method. Anothersection (Section 5.0) details a preferred embodiment consisting of thepipeCURE structure and the C-matrix method. However, all othercombinations are possible and may be preferable depending on thespecific engineering application.

The algorithmic structures (18) are described below in subsequentdescriptive sections as processing input samples in a “batch processing”mode, meaning that sampled data from the sensor signals are grouped intoblocks of samples, which are then processed one block at a time. Theblock size is arbitrary and is not intended to limit the scope of thepresent invention. For example, the block size can be as small as one,in which case, the batch mode reduces trivially to “sample-by-sample”processing.

In some situations, the EGSVs of the source signals vary with time, due,for example, to changing geometric relationships among the locations ofthe sources, the receiving sensor array, or multipath reflectors. Forsuch situations, two methods are presented for determining the initialEGSVs for each block of samples. These methods incorporate a techniqueknown as α-β tracking into the block initialization process and therebymake the acquisition, capture, separation, and recovery of sourcesignals more stable when any of the sources or the receive sensor arrayis moving. Two block initialization methods are presented for thispurpose and are called α-βCURE and μCURE. As indicated by block 19 inFIG. 1, these two methods are applicable to any of the basic algorithmicstructures 18.

FIG. 2A shows the system of the present invention in general form,employing the first iterative method referred to above as thebeamform—cross-cumulant method. The system includes four principalcomponents: a sensor receiving system 20, an estimated generalizedsteering vector (EGSV) computation module 22, a supplemental computationmodule 24 and a linear combiner 26.

The sensor receiving system 20 includes components not shown in FIG. 2A,but which will be discussed in more detail below, including an array ofsensors to convert incident signals into electrical form, and some formof signal conditioning circuitry. In each of the embodiments andapplications of the invention, incident signals 27 are sensed by thesensor receiving system 20 and, after appropriate processing, thesupplemental processing module 24 generates desired output signals orparameters as needed for a specific application of the invention. Forexample, in a signal separation application, the output signals will bethe reconstructed cochannel signals received from separate signalsources. An important aspect and advantage of the invention is that thereceived signals may be of any modulation or type.

A key component of the signal processing system of the invention is theestimated generalized steering vector (EGSV) computation module 22,which receives the conditioned received signals over line 28 from thesensor receiving system 20, and computes an EGSV from the conditionedsignals. The EGSV, in accordance with the invention, results inoptimization of a utility function that depends on fourth or highereven-order statistical cumulants derived from the received signals. Atthis point in the description, cumulants have not yet been defined, butthey are discussed in Sections 4.0 and 21.0. For the present, it issufficient to note that cumulants are fourth-order (or highereven-order) statistical moments of the received signals.

The EGSV computation module 22 functions in an iterative manner incooperation with the linear combiner 26. The linear combiner 26 begins afirst iteration with an initial EGSV, indicated at 30, which may be anestimate or simply a random initial vector quantity. The linear combiner26 then combines the received signals on line 28 with the initial EGSVand generates a single-channel combined signal on line 32 to the EGSVcomputation module 22. The latter performs its cumulant computation andgenerates a revised EGSV, which is fed back to the linear combiner 26over line 34. The iterative process continues until the EGSV hasconverged to an appropriately accurate solution, which is output to thesupplemental computation module 24 over line 36. The output on line 36is also referred to as an estimated generalized steering vector, whichclosely approximates the actual generalized steering vector andrepresents the weighted sum of ordinary steering vectors correspondingto all multipath components of signals from a source. Where no multipathcomponents are present, the generalized steering vector reduces to anordinary steering vector for the source. However, the term “generalized”steering vector is used frequently in this specification, to convey themeaning that the steering vector generated automatically and dynamicallytakes into account the possibility of receiving a signal over multiplepaths.

The function of the supplemental computation module 24 depends on thespecific application of the invention. This module makes use of theconditioned received signals on line 28 and the generalized steeringvector on line 36, and generates the desired output signal or parameterson output line 38.

It should be understood that the components shown in FIG. 2A, and inparticular the EGSV computation module 22 and the linear combiner 26,function to generate a single estimated generalized steering vector online 36, corresponding to a single signal source from which signals arereceived at the sensor receiving system 20. In a number of applications,as will be further described, multiple linear combiners 26 and multipleEGSV computation modules 22 will be needed to separate signals frommultiple sources. Similarly, in most applications the supplementalcomputation module 24 must also be replicated, at least in part, togenerate multiple desired output signals.

Another aspect of FIG. 2A to keep in mind is that it is an imperfectattempt to depict a number of the invention's many possible forms in asingle conceptual diagram. The concept of the generalized steeringvector is central to all forms of the invention. The estimatedgeneralized steering vector (EGSV) provides a means for tracking sourcesthrough time free of discontinuity. The estimated generalized steeringvector computational module, 22 in FIG. 2A, may be partially merged withthe supplemental computation module 24 or the order of computationmodified to accommodate physical implementation. This is true in thecase of a specific implementation of a signal recovery system, describedsubsequently, in which recovered signals are generated as outputs of thesupplemental computation module 24.

An important aspect of the invention is that the EGSV computation module22, or multiple modules in some embodiments, compute estimatedgeneralized steering vectors extremely rapidly, either by directcomputation or by an iterative process that convergessuperexponentially. Moreover, each EGSV represents the weighted sum ofsteering vectors associated with multipath components derived from asingle signal source. As will be described in the next descriptivesubsection, multiple EGSVs can be conveniently processed to reconstructand separate signals from multiple signal sources.

As the description of the various embodiments proceeds, it will becomeapparent that certain components of the invention are common to manyapplications. In terms of the components shown in FIG. 2A, the commoncomponents include the EGSV computation module 22, the linear combiner26 and signal conditioning portions of the sensor receiving system 20.Accordingly, in many instances it would be advantageous to fabricatethese components as a monolithic semiconductor device or chip, to bemounted in close association with other components in the supplementalcomputation module 24, which vary by application. Alternatively, a setof semiconductor chips could include various embodiments of thesupplemental computation module 24, such as for signal separation,direction finding and so forth.

FIG. 2B illustrates the form of the invention that uses the seconditerative method (12, FIG. 1), called the C-matrix method, forgenerating EGSVs. It will be observed that the figure is similar to FIG.2A, except that there is no linear combiner. Instead there is across-cumulant matrix computation module 40 and a structured quadraticform computation module 42.

The cross-cumulant computation module 40 receives conditioned sensorsignals over line 28 from the sensor receiving system 20, computes thecross-cumulants of all combinations of the sensor signals and stores theresults in a matrix having a particular mathematical structure. Thismatrix of cross-cumulants, denoted C, has dimensions P²×P², where P isthe number of signal sources, and is stored for subsequent computations.This C matrix is computed before any iterative computations areperformed, and it will be noted that the EGSVs fed back from the EGSVcomputation module 22 over line 34 are coupled to the structuredquadratic form computation module 42 and not to the cross-cumulantmatrix computation module 40.

The structured quadratic form computation module 42 is used in aniterative computational loop that, starting with an initial estimatedgeneralized steering vector (EGSV) on line 30, produces a series ofsuccessively improved values for the EGSV, until a termination test issatisfied. In each cycle of the iterative loop the module 42 receives aninput EGSV and outputs a cumulant strength function (on line 32), whichis obtained by computing a structured quadratic form involving the Cmatrix and the input EGSV. The mathematical details of the computationare described in a later section. The cumulant strength function, soobtained, is output to the EGSV computation module 22 to update theEGSV. As in FIG. 2A, the nature of the supplemental computation module24 in FIG. 2B depends on the particular application of the invention.

The EGSV computation module 22 together with the structured quadraticform computation module 42 are part of an iterative computational loopthat produces a series of successively improved values for an EGSV of asource signal, starting from an initial value. The EGSV computationmodule 22 receives as input a cumulant strength value for the currentvalue of an EGSV. Based upon this cumulant strength, the module 22determines a new value for the EGSV that will cause the value of thecumulant strength value to increase in absolute value. This new EGSV isoutput to the structured quadratic form computation module, where itreplaces the old EGSV, and the computation cycle is repeated until atermination test is satisfied.

FIG. 2C depicts the direct computation method of the invention inconceptual form using the same basic structure as FIGS. 2A and 2B. TheEGSV computation module in this figure is referred to by numeral 22′,since it performs its function differently from the module 22 in FIGS.2A and 2B. The EGSV computation module 22′ receives as input the Cmatrix computed by the cross-cumulant matrix computation module 40. TheEGSV computation module 22′ computes the solutions to a fourth degreepolynomial equation, from which the EGSVs of one or two sources aredirectly determined. The mathematical details of the particularpolynomial equation are described in a Section 7.0. The values of theEGSVs, so determined, are output to the supplemental computation module24.

It will be understood that FIGS. 2A, 2B and 2C illustrate the processingrequired to recover a single source signal from among a plurality ofpossible source signals, for purposes of explanation. We will now turnto the case of recovery of multiple signals.

2.1 Signal Separation Concept:

The present invention has a number of applications in the communicationsfield and, more specifically, in the separation of cochannel signals.FIG. 3A illustrates the concept of signal separation using the firstiterative method (10, FIG. 1) and differs from FIG. 2A in that there aremultiple linear combiners 26.1 through 26.P and multiple estimatedgeneralized steering vector (EGSV) computation modules 22.1 through22.P. The supplemental computation module 24 has been replaced by asupplemental computation module 24A for signal recovery, the details ofwhich will be explained with reference to FIG. 4A. The sensor receivingsystem 20 outputs the received signals in conditioned form on line 28,which is connected to each of the linear combiners 26 and EGSVcomputation modules 22. Signal conditioning in the sensor receivingsystem, 20, which is described subsequently, may be used to transformthe received signals to P sets of signals, where P is the number ofsignal sources being received, and to be separated. As in FIG. 2A, theEGSV computation modules 22 and the linear combiners 26 cooperate toproduce converged values of estimated generalized steering vectors onoutput lines 28.1 through 28.P. The supplemental computation module 24Auses these generalized steering vectors and the received signals onlines 28 to generate P separate recovered signals on lines 38. How thislatter step is accomplished is best understood from FIG. 4A, which willbe discussed after consideration of FIGS. 3B and 3C.

FIG. 3B depicts a cochannel signal separation and recovery systemsimilar to that of FIG. 3A, but using the second iterative method (12,FIG. 1). As in FIG. 2B, there is a single cross-cumulant matrixcomputation module 40 but, unlike the FIG. 2B system, there are multiple(P) structured quadratic form computation modules 42.1, 42.2 . . . 42.P,each of which receives cross-cumulants from the matrix computationmodule 40 and conditioned input signals from line 28. Cumulant strengthvalues generated by the structured quadratic form computation modules 42are supplied to respective EGSV computation modules 22.1, 22.2 . . .22.P, which output recomputed EGSVs for feedback over lines 34.1, 34.2 .. . 34.P, respectively, to the structured quadratic form computationmodules 42.1, 42.2 . . . 42.P, and converged values of the EGSVs onlines 36.1, 36.2 . . . 36.P.

FIG. 3C depicts a signal separation and recovery system similar to thoseof FIGS. 3A and 3B, but using direct computation instead of an iterativemethod. This figure is also closely similar to FIG. 2C, except that theESGV computation module 22′ generates two EGSVs on output lines 36.1 and36.2, and supplemental processing is performed in the supplementalcomputation module for signal recovery, which generates two recoveredsignals for output on lines 38.1 and 38.2.

As shown in FIG. 4A, the supplemental computation recovery module forsignal recovery 24A includes a recovery beamformer weight vectorcomputation module 44 and multiple recovery beamformers 46.1 through46.P. Computation of the weight vectors in module 44 is made inaccordance with a selected known technique used in signal processing.The beamforming weight vectors for signal recovery are computed directlyfrom the generalized steering vectors. This is done by one of twomethods: (1) by projecting each generalized steering vector into theorthogonal complement of the subspace defined by the span of the vectorsof the other sources by matrix transformation using the Moore-Penrosepseudo-inverse matrix; or (2) by using the Capon beamformer, also calledthe Minimum Variance Distortionless Response (MVDR) beamformer in theacoustics literature. The module 44 generates, from the P inputestimated generalized steering vectors (EGSVs), a set of P weightvectors w₁ through w_(P), on lines 48.1 through 48.P. Implicit in thefunction performed by the module 44 is the orthogonalization of theoutput signals, such that the weight vectors are mutually orthogonal,i.e. each is representative of a separate signal. Alternatively, themodule 44 may be implemented in the manner described in Section 3.0 andSection 21.0.

The weight vectors w₁ through w_(P) are applied to the received,conditioned signals on line 28, in the recovery beamformers 46.1 through46.P. This is basically a process of linear combination, wherein eachsignal component is multiplied by a corresponding component of theweight vector and the results are added together to produce one of therecovered signals.

As will be seen from the more detailed description of the preferredembodiments, the functions of the linear combiners 26 (in the case ofFIG. 3A), the EGSV computation modules 22, the recovery beamformerweight vector computation module 44 and the recovery beamformers 46 maybe combined in various ways. Moreover, all of these functions, togetherwith signal conditioning performed in the sensor receiving system 20,may be conveniently implemented in one or more integrated circuits.

2.2 Signal Separation Concept in the Multipath Environment:

As will be later described in more detail, signal separation inaccordance with this invention has important advantages in the contextof multipath signal processing. Communication signals, particularly inan urban environment, often reach a receiver antenna over multiplepaths, after reflection from geographical features, buildings and otherstructures. The multiple signals arrive at the antenna with differentsignal strengths and subject to relative time delays and other forms ofdistortion. Because they arrive from different directions, such signalsmay be separated in a conventional cochannel signal separation system.However, the cochannel signal processing system of the present inventionwill automatically combine the multipath components if they are stillcoherent with each other. Coherency, in this context, is a relative termthat simply indicates the degree to which the signals are identical.Multiple signals are classified as coherent if they are relativelysimilar to each other, as measured by a cross-correlation function overa finite time interval. Multipath components are noncoherent, andtherefore not combinable, when they suffer large relative time delays orwhen a signal transmitter or receiver is in motion, causing a Dopplershift in the transmitted signal that affects one path more than another.

The key to successful processing of multipath components is that eachgeneralized steering vector referred to with reference to FIGS. 1, 2A-2Cand 3A-3C corresponds to the sum of all of the mutually coherentmultipath components of a signal source incident on the sensor receivingsystem 20. The generalized steering vectors from all signal sources arethen converted to a set of beamforming weight vectors (in recoverybeamformer weight vector computation module 44), without the need forknowledge of array geometry or array manifold calibration data (whichrelates array steering vectors with angles of arrival for a particulararray). For many of its applications, the invention is, therefore,completely “blind” to the array manifold calibration data and thephysical parameters of the array.

2.3 Direction Finding Concept:

The invention as described with reference to FIGS. 2A-2C, 3A-3C and 4Aprovides for cochannel signal separation, even in a multipathenvironment, without use of specific sensor array manifold information.The array parameters, and specifically array calibration data thatrelate array steering vectors with directions of signal arrival, areneeded if one needs to know the angular directions from which signalsare being received. FIG. 4B illustrates this concept for a single signalsource referred to as source k. Although the concept depicted in FIG. 4Bis conventional, the manner in which the present invention generates theinput steering vectors is novel. A generalized steering vector, referredto as a_(k), is received from one of the EGSV computation modules 22.k(FIG. 3A) on line 36.k and is processed in a direction of arrival (DOA)search module 24B, which is a specific form of the supplementalcomputation module 14 (FIGS. 2A-2C). The DOA search module 14B usesstored array calibration data 50, which associates each possibledirection of arrival with a steering vector. These calibration data aretypically stored in a memory device as a lookup table. In the DOA searchmodule 24B, a reverse table lookup is performed to obtain the twoclosest directions of arrival from the input steering vector, asindicated in block 52. Then a more precise angle of arrival is obtainedby performing an interpolation between the two angle values retrievedfrom the calibration data 50, as indicated in block 54. The direction ofarrival (DOA) parameter for the k^(th) signal source is output fromcomputation module 24B on line 38.k.

2.4 Transmitter/Receiver Concept:

As will be discussed further in section 12.0, an important applicationof the invention is to two-way communication systems. In manycommunication systems, the allocation of transmission frequencies withina geographical area, such as in a predefined cell in a cellulartelephone system, is often a limiting factor that determines the maximumnumber of active users that the system can handle. In accordance withthis aspect of the present invention, information derived from signalsreceived and separated in a receive mode of operation are used togenerate signals to a transmit antenna array, in such a way thatseparate information signals can be transmitted to respective remotestations using the same frequency.

As already discussed, the present invention makes use of an array ofsensors or antennas to separate received signals of the same frequency(cochannel signals). In a two-way communication system of the typehaving multiple mobile units, it would be impractical, in general, torequire the use of an antenna array at each remote or mobiletransmitter/receiver. These remote units may be installed in vehicles orbe hand-held devices for which the use of an antenna array is eitherinconvenient or simply impossible. However, in most communicationsystems, the communication path between remote units is completedthrough one or more base stations operating as a receiver andtransmitter. Since the base stations are generally larger and morepowerful than the mobile units and are fixed in location, they may beconveniently structured to include antenna arrays for both receiving andtransmitting. As discussed above, a receive array connected to thesystem of the invention provides for separation of received cochannelsignals. Moreover, an important by-product of the signal separationprocess is a set of generalized steering vectors, each associated with aseparate signal source.

As shown in FIG. 5 for a single remote transmitter/receiver station,received signals are recovered on line 38, in the manner discussed withreference to FIG. 2A. In a two-way communication system, a transmitter56 uses the generalized steering vector corresponding to the receivedsignal source, obtained from the EGSV computation module 22, in order togenerate a weight vector for application to a transmit antenna array(not shown). The transmitter 56 also receives, on line 58, aninformation signal to be transmitted. Typically, this will be in theform of a digitized voice signal, although it may be a data signal ofsome other type. The transmitter 56 generates the transmit weight vectorin accordance with a technique to be described below in Section 12.0,modifies the information signal in accordance with the weight vector,modulates a carrier signal with the weighted information signalcomponents, and sends a set of signals to the transmit array, asindicated by lines 60. The carrier frequency, although the same for eachof the transmitted cochannel signals, is usually selected to have afrequency different from the receive signal frequency. As will also befurther discussed below, in the more general case of multiple receivedsignals and multiple transmitted signals, the transmitter 56 generatesmultiple weight vectors, which are applied to the respective informationsignals to be transmitted, then linearly combined, antenna element byantenna element, and finally modulated to produce a set of compositeantenna element signals for coupling to the transmit antenna array.

It will be appreciated that this aspect of the invention provides asimple but effective technique for use in a communication system basestation, for receiving cochannel signals from, and transmittingcochannel signals to, multiple remote stations in close proximity to abase station and to each other. Limited only by the number of elementsin the receive and transmit arrays, the technique allows for re-use ofthe same frequencies in the multiple remote stations, with a resultantincrease in system capacity or user density, e.g. the number of usersper frequency per unit area (per cell or sector thereof).

2.5 Concept of Separation of Signals in a “Waveguide”:

Up to this point in the description, it has been tacitly assumed thatthe “incident signals” 27 shown in FIGS. 2A-2C and 3A-3C are signalstransmitted through space, the atmosphere, the ocean or some otherrelatively unbounded medium. As will be described in more detail indescriptions of various embodiments of the invention, cochannel signalstransmitted on a waveguide of some type may also be received andprocessed in accordance with the principles of the invention. The termwaveguide is used in quotation marks in the heading of this descriptivesubsection because the term is not intended to be limited to a waveguideoperating at microwave frequencies, or to an optical waveguide in theform of an optical fiber or planar optical waveguide. Instead, the wordwaveguide as used in this specification is intended to cover any ofvarious types of bounded transmission media, including microwavewaveguides, optical waveguides, coaxial cables, or even twisted pairs ofconductors operating at relatively low frequencies.

A common attribute of these “waveguides” is that multiple signals may betransmitted along them using the same frequency but different modes oftransmission, such as different polarization modes for microwave oroptical waveguides, or, in the case of twisted-pair conductors,different signals being applied to different wire-to-groundcombinations. For various reasons, however, the propagation modes maybecome scrambled in the propagation medium and may become difficult orimpossible to separate in a receiver. As shown in FIG. 6, a system inaccordance with the invention can be usefully employed to separate suchsignals. Except for the sensor receiving system 20, the system of FIG. 6is identical with the FIG. 3A system for separating received signals.The “incident signals” 27 are received from a signal “waveguide” 62, asdefined in the preceding paragraph, and are sensed by “waveguide”sensors 52. In the case of a microwave or optical waveguide, the sensors64 take the form of electromagnetic probes or optical sensorsappropriately inserted into the waveguides. For twisted-pair conductors,the sensors may include appropriate circuitry connected to theconductors and ground. Signals from the sensors 64 are subject toreceived signal conditioning, as indicated in block 66, and are theninput to the signal separation system of the invention in the same wayas signals from multiple elements of an antenna array. The systemrecovers the original signals as indicated on lines 38.

Another application of the invention is similar in some respects to therecovery of signals from a bounded waveguide. In magnetic recordingsystems using a high density recording medium in which recording tracksare positioned very close to each other, there is always the potentialfor crosstalk between the signals on adjacent parallel tracks.Maintaining an acceptably low level of crosstalk imposes a limitation onthe proximity of the tracks and the overall recording density. In thisembodiment of the invention, a higher level of crosstalk can betolerated because signals retrieved from adjacent tracks can beseparated using the signal recovery system of the invention. In thiscase, the “waveguide” sensors 64 are adjacent playback sensors in amagnetic recording apparatus. There is no “waveguide” as such; nor arethe signals transmitted through an unbounded medium. Instead they aresensed electromagnetically from a recording medium on a moving magnetictape or disk.

2.6 Preview of Iterative Embodiments to be Described:

As discussed in more general terms above, there are two alternativeiterative methods that may be used in accordance with the invention inthe context of separation and recovery of cochannel signals. Beforeproceeding with the detailed descriptions of those methods in terms ofspecific embodiments, it may be helpful to provide a an overview of theiterative methods at a different level of abstraction from that of thepreceding figures. FIG. 7 provides the basis for this overview. Some ofthe technical terms used in FIG. 7 are introduced for the first time inthis specification and may not be completely clear until the completedescription is studied.

As shown in FIG. 7, the signal separation and recovery process involvesa number of manipulations of the estimated generalized steering vectors(EGSVs) pertaining to the multiple signal sources. In block 70 EGSVinitialization is performed. This is simply the selection of initialEGSV values (on line 30 in FIGS. 2A and 2B) from which to beginprocessing. It will be recalled that processing is performed in a batchmode in which sequential blocks of data are processed. The initial EGSVsmay be estimates carried forward from an already processed previousblock of data, or they may be generated from scratch, using eitherrandom quantities or a cumulant eigen-decomposition algorithm.

Another function performed in block 70 is to “project” the initial EGSVsinto a P-dimensional signal space, where P is the number of signalsources. The antenna array provides sets of input data signals that areM-dimensional, where M is the number of elements in the array.Throughout the computations performed in accordance with the invention,there is often a design choice to be made because the mathematicalmanipulations may be performed on these M-dimensional vector quantities,or on corresponding P-dimensional quantities, where P is the number ofsignal sources. Ultimately, the recovered signals are output as Pone-dimensional signals, since there is one recovered signal per source,but signal recovery requires beamforming in the sensor space. Thetransformation from M-dimensional sensor space to P-dimensional signalspace is called “projection,” and the reverse transformation, fromP-dimensional signal space to M-dimensional sensor space, is called“backprojection.”

Block 72 in FIG. 7 describes EGSV prioritization. This aspect of theinvention has not yet been discussed but, simply stated, prioritizationis needed to provide a rational basis for choosing which of multiplesignals should be recovered first. The sensor array, having M elements,inherently limits the number of cochannel signals that may be recoveredto M. If more than M signal sources are actively transmitting, the firstM signals are selected on the basis of their non-Gaussianity, asdetermined by either of two methods: using the C-matrix or beamformingand computing cross-cumulants. The resulting priority list of sources ispassed to a signal separation iteration block 74, which uses one of thetwo iterative methods to obtain convergence for each EGSV in turn,starting with the highest priority source. The steps of the iterativeprocedure include updating the EGSV using either the C-matrix or thebeamforming and cross-cumulant computations, then using a conventionaltechnique, such as the Gram-Schmidt procedure, to ensure that each EGSVis orthogonal to already-processed higher-priority EGSVs. These stepsare repeated until convergence is achieved for each signal source.

Block 76 of FIG. 7 describes another ,practical issue in signal recoverysystems that use batch processing. Each “port” from which a recoveredsignal is output in processing a block of data must be correctlyassociated with a recovered signal from the previous data block. Thisassociation is performed by comparing the EGSVs. The ports cannot beassociated merely on their positions in the priority list because signalsources may come and go from list as time passes.

Another batch processing issue is addressed in block 78 of FIG. 7. Thephase angle of an EGSV generated in processing one block of data may notexactly match the phase angle as determined in the next block. Thisprocessing step applies an EGSV phase adjustment to eliminate anydiscontinuity from block to block.

Next, in block 80 of FIG. 6, the EGSVs are “backprojected” from theP-dimensional signal subspace to the M-dimensional sensor space andthen, in block 82, the backprojected EGSVs are used to beamform andrecover the output port signals.

This overview provides an introduction into the various embodiments andforms that the invention may take. The foregoing and other aspects ofthe invention will now be discussed in more detail.

3.0 Preferred Embodiment Using eCURE:

As discussed more generally above, the present invention pertains to asystem and method for processing and recovering cochannel signalsreceived at a sensor array. In this descriptive section, a practicalembodiment of the cumulant recovery (CURE) system is disclosed. Becausethis embodiment uses eigenvectors and eigenvalues in part of itscomputation, it is referred to as the eigenCURE system, or simply theeCURE system. The embodiment disclosed uses the first iterative method(the beamform and cross-cumulant method, first introduced at 10 in FIG.1). It will be understood, however, that the eCURE system may bemodified to use the C-matrix iterative method (12, FIG. 1).

3.1 Overview And System Hardware:

FIG. 8A shows the basic components of the eCURE system, including asensor array, indicated by reference numeral 110, which receives signalsfrom various directions, as indicated at 112, a bank of receivers 114,and a sampler and analog-to-digital converter bank 116. The separatesignals from the elements of the sensor array 110 are coupled directlyto the receiver bank 114, which performs conventional filtering andfrequency downconversion functions. The sensor signals are then sampledat a high rate and converted to digital form in the sampler andanalog-to-digital converter bank 16. At this point, the signals havebeen filtered, downconverted and digitized and processing is about tobegin. It can be appreciated that, because each sensor in the array 110provides a stream of digitized signals, processing may be convenientlyperformed in batches or blocks of data. From time to time, referencewill be made in this description to current and previous data blocks.The block size is critical only in the sense that the size selectedaffects processing speed and accuracy of estimation.

The first major processing step is preprocessing the sensor data, whichis performed in a preprocessing computer 120. As will be discussed inmore detail with reference to FIG. 10, the preprocessing computer 120performs four important functions:

It “whitens” the directional components of the signals using a techniqueknown as eigendecomposition, which will be discussed further below.

It estimates the number of signal sources being received. P is thenumber of sources and P_(e) is the estimated number of sources.

It reduces the dimensionality of the sensor data from M, the number ofsensors, to P_(e), the estimated number signal sources.

It scales the numerical values of the signals to normalize the powers ofthe sources, permitting weak signals to be separated in addition tostronger ones.

The other major components of the cochannel signal recovery system are asignal recovery controller 122, multiple signal extraction ports 124.1through 124.L, an orthogonalizer 126, and multiple demodulators 128.Preprocessed sensor signals are transmitted over lines 130 in parallelto each signal extraction ports 124.1. Each port in this embodiment ofthe invention is implemented as a separate computer processor. Using theiterative technique described above with reference to FIGS. 2A and 3A,each of the signal extraction ports 124 generates output signals derivedfrom a separate source. These signals are output on lines 132 to thedemodulators 128, which produce usable output signals on lines 134. Ifthe information contained in the signals is audio information, theoutputs on lines 134 may be connected to separate loudspeakers or otheraudio processing equipment (not shown). The function of theorthogonalizer 126 is to ensure that each of the ports 124 is associatewith a separate signal source. The signal recovery controller 122performs various control functions in conjunction with the signalextraction ports 124 and the orthogonalizer 126. The controller 122receives the source count estimate P_(e) from the preprocessing computer120 over line 136 and also receives eigenstructure parameters from thepreprocessing computer over line 138. The latter are also transmitted tothe signal extraction ports 124, and the source count estimate istransmitted to the orthogonalizer 126. The controller 122 also sends apriority list to the orthogonalizer 126 over line 140. Finally, thecontroller 122 sends adaptation flags to the signal extraction ports 124over lines 142 and receives capture strength values from the signalextraction ports over lines 144. The specific functions of these signalswill become apparent as the description proceeds. Basically, onefunction of the controller 122 is to keep track of signal sources asthey appear and disappear and to make sure that the signal extractionports 24 and the orthogonalizer 126 handle appearing and disappearingsignal sources in an appropriate manner.

In the general terms used in FIG. 3A, the sensor receiving system 20includes the sensor array 110, the receiving bank 114, the sampler andanalog-to-digital converter 116 and the preprocessing computer 120. Thefunctions performed by the linear combiners 26, the EGSV computationmodules 22 and the supplemental computation module for signal recovery24A, are performed in the signal recovery controller 122, the signalextraction ports 124 and the orthogonalizer 126. Because this practicalembodiment must maintain association of signal sources to physicaloutput ports, and must be able to adapt dynamically to the appearanceand disappearance of signal sources, the architecture of the system isnecessarily different from the conceptual architecture of FIGS. 2A, 3Aand 4A. The fundamental functions performed in the system of FIG. 8Aare, however, the same as those described with reference to the earlierfigures.

FIG. 8B is a hardware block diagram of one implementation of the eCUREsystem of the present invention, as used to separate multiple signalscontaining audio information. Identical reference numbers have been usedin this figure to refer to components that appear in both FIG. 8A andFIG. 8B. The preprocessor computer 120 is implemented as a separatecircuit card. It is of little significance whether the preprocessorcomputer is implemented on a single circuit card or in a stand-alonecomputer. The functions performed in both cases would be identical.Similarly, the controller 122 and orthogonalizer 126 are implemented inone computer on a single circuit card, as indicated in FIG. 8B. Thesignal extraction ports 124, one of which is shown, are implemented onseparate computers on circuit cards and the demodulators 128 are alsoimplemented on a separate computer. All of the computers mentioned abovemay be of any appropriate type, but in a presently preferreddemonstration embodiment, they are Intel model i860 processors. Thecomputers are connected to a high-speed crossbar switching network 150,such as Part No. ILK-4, from Mercury Computer Systems, Lowell, Mass.,01854.

The sensors 110 may be of any appropriate type, such as Part No.10-183-244, from TRW Inc., Sunnyvale, Calif., 94088. The receivers 114in this embodiment include a VME synthesizer module (Part No. 1600MSYN-5, from APCOM Inc., Gaithersburg, Md., 20878), a local oscillatormodule (Part No. 1600M LOD-1, also from APCOM Inc.) and a VME RFconverter module (Part No. 1600M RFC-5, also from APCOM Inc.). Thedigitizer bank 116 may include an Access256 motherboard (Part No.MOB256-4) and analog input and digital output cards from CeleritySystems, Inc., San Jose, Calif., 95117.

The receiver bank 114 tunes each sensor to the desired frequency anddownconverts any signals received. One receiver is allocated to eachsensor. The receivers 114 enable the system to isolate a singlefrequency of interest and to translate it to a frequency where it can bemore conveniently digitized and processed. No demodulation is performedat this point. For example, one embodiment of the invention hasreceivers that downconvert the received signals to a center frequency of225 kHz and a bandwidth of 100 kHz (specifically the 3 dB bandwidth).

The digitizer bank 116 converts the received signals to digital samples.Only real samples, not complex samples, are generated at this stage. Inthe illustrative system, the digitizer bank 116 consists of a digitizermotherboard and an input daughtercard that samples up to eight channelssimultaneously at a rate of up to 10 megasamples per second. The samplesare exported for further processing through an input card 151. The dataprocessing system performs digital filtering of the input data, with thedigital filtering card 152, and converts the real samples to complexvalues needed for processing, using a real-to-complex conversion card153.

All of the processor cards mentioned above are connected to thehigh-speed crossbar switching network 150. A system manager 154 in theform of a Model 68040 CPU (Part No. CPU-40B/16-02, from Force ComputersInc., San Jose, Calif., 95124) controls this demonstration system, withoperator interface being provided by a workstation 56, such as aPowerlite notebook workstation (Part No. 1024-520-32, from RDI ComputerCorporation, San Diego, Calif., 92008). In this hardware architecture,the demodulated signal outputs are connected to loudspeakers 157 and ahard disk drive 158 is provided for storage of received messages. Thesystem manager tunes the receivers 116, instructs the digitizer torecord samples, configures and initiates operation of the dataprocessing system, and controls a peripheral interface board (Part No.MVME 162-63, from Motorola Computer Group, Tempe, Ariz., 85282), throughwhich communication is had with the loudspeakers and the hard disk. Itmust be understood that, because this is a demonstration system, many ofthese components would not be needed in some implementations of theinvention. As noted earlier, many of the components could beconveniently implemented in the form of a single integrate-circuit chip,or multiple chips.

FIG. 9, which is broken into two figures designated FIGS. 9A and 9B, isa block diagram similar to FIG. 8A but showing in each block more detailof the functions performed by each component of the system. Thefunctions will be described with reference to the more detailed diagramsthat follow. Another feature shown in FIG. 9 but not in FIG. 8A is theoutput of generalized steering vectors on lines 170. The generalizedsteering vectors, as discussed earlier in this specification, are animportant product of the signal recovery process, along with therecovered signal outputs. The generalized steering vectors are shown asbeing used in a DF search processor 172, which generates signal sourcebearings or directions of arrival (DOA), output on lines 174, consistentwith the description with reference to FIG. 4B.

3.2 Preprocessing:

The functions performed by the preprocessing computer 120 are shown indetail in FIG. 10. The preprocessor performs a block-by-block analysisof “snapshots” of data from the sensor array, which has M elements. Thepreprocessor determines the eigenvectors, the number of signal sourcesP, and the signal subspace of the received array measurement data. Thepreprocessor also filters the received array data, transforming it fromM-dimensional sensor space to P-dimensional signal subspace. Thistransformation also renders the steering vectors of the transformedsources orthogonal to each other, which greatly accelerates convergenceon recovered signal solutions later in the processing, and thetransformed source powers are made equal.

A received signal that satisfies the narrow-band assumption (to bedefined below) can be described by the following equation:

r(t)=As(t)+n(t),

where r(t) denotes the array measurements collected by M sensors, n(t)is the measurement noise, A is the steering matrix that models theresponses of the sensors to the directional sources, and s(t) is atime-varying signal. The sampled sensor signals, represented by afunction of time r(t), are input to the preprocessing computer until acomplete block of data has been received. While the block is beingfurther processed, another block of data is input to a buffer in thecomputer (not shown). As indicated at 180, each block of data is firstsubject to computation of an array covariance matrix for the currentblock of data. The sample covariance matrix {circumflex over (R)} from Nsamples or “snapshots” of data is given by:$\hat{R} = {\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}{{r(t)}{{r^{H}(t)}.}}}}$

As indicated at 182, a further important step in preprocessing iseigendecomposition, which decomposes the covariance matrix as:

{circumflex over (R)}=EΛE^(H), E^(H)E=I_(M),

where the diagonal matrix Λ=diag(λ₁, λ₂, . . . λ_(M)) contains theeigenvalues of {circumflex over (R)} (which are positive) in adescending order, and the columns of the matrix E consists of thecorresponding eigenvectors. In this description, the subscript appendedto the identity matrix I indicates its size. In the case of truestatistics and with fewer sources than sensors (P<M), the last (M-P),eigenvalues are identical and they are equal to the noise varianceλ_(M-P+1)=. . .=λ_(M)=σ². With sample statistics, the last (M-P)eigenvalues are different with a probability of one, and a statisticaltest has to be performed to determine the number of sources P.

In the presently preferred embodiment of the invention, thepreprocessing computer uses a combination of two estimates of the numberof sources, as indicated in block 184: the estimate determined byAkaike's Information Criteria (AIC) and the estimate determined byRissanen's Minimum Description Length principle (MDL). Specifically, thepreprocessing computer averages the AIC and MDL functions and finds thesingle maximizer of the average. The equations for making thisestimation are given, for example, in a dissertation by Mati Waxsubmitted to Stanford University in March 1985, and entitled “Detectionand Estimation of Superimposed Signals,” and in particular thesubsection headed “Estimating the Number of Signals,” beginning on page69 of the paper. AIC generally overestimates the number of sources, andMDL generally underestimates the number of sources. By averaging the twoestimates, a good result is obtained. The averaged estimate of thenumber of sources is given as the minimizer of the cost function:$\begin{matrix}{P_{e} = \quad {\underset{0 \leq k < M}{argmin}\left\{ {{N\quad {\log \left\lbrack {\left( {\frac{1}{M - k}{\sum\limits_{i = {k + 1}}^{M}\lambda_{i}}} \right)^{M - k}/{\coprod\limits_{i = {k + 1}}^{M}\quad \lambda_{i}}} \right\rbrack}} +} \right.}} \\{\left. \quad {\frac{1}{2}\quad {k\left( {{2M} - k} \right)}\left( {1 + {\frac{1}{2}\quad \log \quad (N)}} \right.} \right\}.}\end{matrix}$

After the number of sources is estimated, the preprocessor computercomputes estimates of the eigenvectors and eigenvalues of the signal andnoise subspaces:${E = \left\lbrack {E_{s},E_{n}} \right\rbrack},{\Lambda = \left\lbrack {\Lambda_{s},{{\hat{\sigma}}^{2}I_{M - P_{e}}}} \right\rbrack},{{\hat{\sigma}}^{2} = {\frac{1}{M - P_{e}}{\sum\limits_{k = {P_{e} + 1}}^{M}{\lambda_{k}.}}}}$

E_(s) and E_(n) are estimates of signal and noise subspace eigenvectors,respectively, and {circumflex over (σ)}_(n) ² is an estimate of thenoise power. The diagonal matrix Λ_(s) contains the estimates of signalsubspace eigenvalues. Once the subspaces are found, the preprocessorcomputer determines the transformation matrix T, as indicated in block186, from:

T=E_(s)(Λ_(s)−{circumflex over (σ)}_(n) ²I)^(−½)

The transformation T is then applied to the sampled measurements r(t),as indicated in block 188:

y(t)=T^(H) r(t)=T^(H) As(t)+T^(H) n(t)=y _(s)(t)+T^(H) n(t),

where y_(s)(t) denotes the signal component in y(t). It can be provedthat the covariance matrix of y_(s)(t) is the identity matrix. Thisindicates that the steering vectors of the sources after transformationto the P_(e)-dimensional subspace are orthogonal to each other and thesource powers are equalized.

The transformation reduces the dimensionality of the sensor measurementsfrom M, the number of sensors, to P_(e), the estimated number of signalsources. The eigendecomposition performed in the preprocessing computeris a well known technique, sometimes referred to as spatialprewhitening, originally developed for use in passive sonar signalprocessing. It is described in more detail in a number of texts onsignal processing. For example, see “Detection of Signals in Noise,” byAnthony D. Whalen, Academic Press, New York (1971), beginning at page392. In spatial prewhitening, noise components at each sensor areassumed to correlated (i.e., not completely random noise). Theprewhitening process operates on the noise signals to render themuncorrelated (i.e,. “whitened”). An alternative to eigendecomposition isto use covariance inversion in preprocessing. In effect, the latterprocess whitens both noise and signal components of the sensor signalsand it can be at least intuitively understood that this is a lessdesirable approach since it renders the signals less easy to separatefrom the noise than if spatial prewhitening of the signal only wereused. However, the present invention will still separate cochannelsignals using covariance inversion instead of eigendecomposition as apreprocessing step. Currently, eigendecomposition, or spatialprewhitening, is the preferred approach for preprocessing.

3.3 Operation of an Active Signal Extraction Port:

A single active signal extraction port, one of the L ports 124 shown inFIG. 8A, is shown in detail in FIG. 11, which extends over two sheets,as FIGS. 11A and 11B. L is the number of physical ports allocated forsignal extraction and is less than M, the number of sensors. Forgenerality, the illustrated port is referred to as the kth port 124.k,and signals pertaining specifically to this port are referred to bysignal names that include the prefix k. An “active” port is one that hasbeen assigned responsibility for separating a signal from the receivedsignal data. The functions performed in an active port are critical tothe invention and are illustrated in FIG. 11.

The active signal extraction port 124.k receives as inputs over line 130the preprocessed sensor data y(t). As in the preprocessing computer 120(FIG. 7), the sensor signals are described as being processed in blocks,although it will be understood that an alternative embodiment of theinvention could be devised to process the signals continuously. Theprincipal output from the port is a recovered signal, designatedg_(k)(t). An important intermediate output is a vector quantity calledthe normalized cumulant vector, referred to as b_(k). Another importantoutput is the generalized steering vector a_(k), which defines thedirectional location of the signal source.

To highlight the (block) iterative nature of the algorithm, thequantities related to the mth analysis block are identified in thisdescriptive section by using the additional index m. For example, thesteering vector estimate provided by the kth port after the mth block isprocessed is denoted as a_(k)(m). The time variable t spans from thestart of the mth block to the end of the mth block, and, with Nsnapshots per block, can be expressed at t ε[(m−1)N+1,mN]. Thequantities E, Λ and T are obviously obtained by processing the mth blockof data, so, for simplicity, they are not written with the (m) index.

It is both logical and convenient to begin describing operation of thesignal extraction port at the point in time when the port has just beenmade active. No accurate signal has been recovered and no cumulantvector has been computed since the port became active. The block numberbeing of input data is examined, as indicated at 200, to determinewhether the current block is the first block. If so, the steering vectora_(k)(0) is set to an initial random vector with a unit norm. As shownin processing block 102, the steering vector a_(k)(m−1) is firstprojected to the signal subspace by transforming it into a value v_(k),using the transformation T, which is input to block 202 as shown. Thesteering vector projected into the signal subspace is computed as:

v _(k)=T^(H) a _(k)(m−1).

These computed values are passed to a beamformer 204, which recovers anauxiliary waveform using v_(k) as weights:

u_(k)(t)=v_(k) ^(H)y(t)/ || v_(k) ||.

The scaling by the norm of v_(k) is performed to ensure numericalstability in the cumulant computations. At convergence, the norm ofv_(k) should be unity (because of the preprocessing performed on themeasurements). The auxiliary source waveform is provided to the cumulantvector computer 206, which computes sample estimates of thecross-cumulant vector b_(k), which has components defined by:

(b_(k))_(i)=cum(u_(k)(t),u_(k) ^(*)(t),u_(k) ^(*)(t),y_(i)(t)),1≦i≦P_(e),

where (b_(k))_(i) denotes the ith component of b_(k). The asteriskindicates that the conjugate of the process u_(k)(t) is used in two ofthe four quantities of which the cumulant is computed. The cumulantvector computer computes a fourth-order cumulant vector of the inputsignals. More specifically, the vector is the cumulant of fourquantities derived from the input signals. An introduction to cumulantsand their properties is provided in Section 20.4 to this specification.

After the cumulant vector b_(k) has been computed in the cumulant vectorcomputer 106, the capture strength c_(k) is computed in a capturestrength computer 208.

The capture strength computer 208 determines the convergence conditionof a port by evaluating the degree of non-Gaussianity of the auxiliarysignal u_(k)(t), and the amount of change between b_(k) and v_(k). Atconvergence, these two vectors should be pointing at the same direction.The non-Gaussianity of the auxiliary signal can be determined from theratio of its fourth-order cumulant to its squared power:$\xi_{k} = \frac{{{cum}\left( {{u_{k}(t)},{u_{k}^{*}(t)},{u_{k}^{*}(t)},{u_{k}(t)}} \right)}}{\left( {E\left\{ {{u_{k}(t)}{u_{k}^{*}(t)}} \right\}} \right)^{2}}$

Using cumulant properties, we obtain $\begin{matrix}{\xi_{k} = \frac{{{cum}\left( {{u_{k}(t)},{u_{k}^{*}(t)},{u_{k}^{*}(t)},{v_{k}^{H}{{y(t)}/{v_{k}}}}} \right)}}{\left( {E\left\{ {{u_{k}(t)}{u_{k}^{*}(t)}} \right\}} \right)^{2}}} \\{= \frac{{{v_{k}^{H}b_{k}}}/{v_{k}}}{\left( {E\left\{ {{u_{k}(t)}{u_{k}^{*}(t)}} \right\}} \right)^{2}}}\end{matrix}$

The denominator can be computed from the auxiliary signal u_(k)(t).Since the covariance matrix of the signal component of y(t) is theidentity matrix, the denominator can be ignored when the signal to noiseratios (SNRs) are high enough. The similarity between b_(k) and v_(k)can be computed from the following:$\eta_{k} = {\frac{{v_{k}^{H}b_{k}}}{{v_{k}}{b_{k}}}.}$

The capture strength c_(k) can be determined from ξ_(k) and η_(k). Oneway is to let c_(k)=ξ_(k)·η_(k). Alternatively, we can set c_(k)=ξ_(k)or c_(k)=η_(k).

The capture strengths c_(k) are provided to the controller unit forpriority determination. The cross-cumulant vector is normalized by itsnorm and fed to the orthogonalizer:

(b_(k)(m)=b_(k/||b) _(k)||).

If the active ports capture different sources, their cross-cumulantvectors should be orthogonal. To force the active ports to capturedifferent sources, the orthogonalization unit uses the Gram-Schmidtprocedure and outputs orthogonalized cumulant vectors (d vectors) to theports. The steering vectors are determined from the orthogonalizedcumulant vectors according to

a _(k)(m)=E_(s)(Λ_(s)−{circumflex over (σ)}²I_(P) _(e) )^(½) d_(k)(m)=T(Λ_(s)−{circumflex over (σ)}²I_(P) _(e) )d _(k)(m).

The steering vectors are provided to the controller unit in order todetermine the port that loses its signal in the event of a sourcedrop-out or to determine which port will be activated in the case of anew signal. The steering vectors can also be used by an optional DFsearch unit to determine the source bearings.

Finally, the active port determines the signal waveform from the sourceit is tracking. However, it is necessary for an active port to maintaingain and phase continuity of its recovered signal at block transitionsin order to prevent block-to-block gain and phase modulation of therecovered signal. To accomplish this goal, we need to examine theproperties of the algorithm in more detail. The algorithm normalizes thesource waveforms to have unit variance and estimates the steeringvectors based on this normalization, i.e., eCURE views the measurementsas:

r(t)=(AΣ_(ss) ^(½)D)(D^(*)Σ_(ss) ^(−½) s(t))+n(t),

where Σ_(ss) is the covariance matrix of the directional sources whichis diagonal since the sources are independent (Σ_(ss)=E{s(t)s^(H)(t)}).The diagonal matrix D contains arbitrary phase factors associated withthe blindness of the steering vector estimation procedure. Even when thesources are stationary, there can be gain (due to Σ_(ss)) and phasemodulations (due to D) on the steering vectors estimates and waveformestimates.

There are two different ways to determine gain and phase modulations foreach block. We can compare the steering vector estimates a_(k)(m) anda_(k)(m−1), which should be pointing to the identical direction atconvergence. Suppose, due to power changes and arbitrary phaserotations, the following relationship holds between the steeringvectors:

a _(k)(m)≈q _(k) a _(k)(m−1)

where q_(k) accounts for the gain and phase factor between the twosteering vector estimates. We can estimate q_(k) using least-squares:

{circumflex over (q)} _(k) =a _(k) ^(H)(m−1)a _(k)(m−1)/||a _(k)(m−1)||²

Using

a _(k)(m)=E_(s)(Λ_(s)−{circumflex over (σ)}²I)^(½) d _(k)(m)

and

a _(k)(m−1)=E_(s)(Λ_(s)−{circumflex over (σ)}²I)^(½) v _(k)

we obtain an alternative way to compute q_(k):

{circumflex over (q)} _(k) =v _(k) ^(H)(Λ_(s)−{circumflex over (σ)}²I)d_(k)(m)/(v _(k) ^(H)(Λ_(s)−{circumflex over (σ)}²I)v _(k)).

In the mth block, the kth block scales the steering vector estimate byq_(k), relative to the previous block. Therefore, it scales the waveformestimate at the mth block by the reciprocal of this quantity. Hence weneed to multiply the waveform estimate by q_(k) (or by its estimate) inorder to undo the scaling done by the processor, which is described asbelow:

g_(k)(t)={circumflex over (q)}_(k)(d_(k) ^(H)y(t))=({circumflex over(q)}_(k) ^(*)d_(k))^(H)y(t)=w_(k) ^(H)y(t)

The second way to compute q_(k) is to force the first component of thesteering vector estimate to be unity. In this process, we simply letq_(k) be the first component of a_(k)(m). After q_(k) is determined,w_(k) will be determined using the orthogonalized cumulant vectord_(k)(m) and q_(k).

Once a source waveform is recovered, it is available for subsequentprocessing as desired. It can be recorded or demodulated and listened towith headphones or loudspeakers.

3.4 The Signal Recovery Controller:

Now that the basic signal extraction method has been described andbefore proceeding to a description of the orthogonalizer function, it islogical to consider next how the signal recovery controller 122 (FIG.8A) operates because this affects operation of the orthogonalizer 126and the signal extraction ports 124. As briefly discussed with referenceto FIG. 9, an important function of the controller 122 is to detectchanges in the status (ON or OFF) of signal sources and to identify lostsources. In addition, the controller 122 maintains a priority list ofports and a related set of adaptation flags that indicate which portsare active.

As shown in FIG. 13, which is spread over three pages as FIGS. 13A, 13Band 13C,, the functions of the controller 122 include logic to detectchanges in the number of signal sources, indicated by block 220, portallocation logic 222, priority list determination logic 224, andadaptation flag logic 226. The logic 220 to detect changes in the numberof sources assumes that there is no more than one change in the numberof sources from one data block to the next. The logic receives theestimated number of signals P_(e) from preprocessing computer 120 andcompares the P_(e) of the previous block with the P_(e) of the currentblock. The results of the comparison determine the value of sourcechange flag, referred to simply as “Flag.” Flag is set to zero for theinitial block. There are three possible outcomes of the comparison forsubsequent blocks of data:

1. If current P_(e)=previous P_(e), Flag=0

2. If current P_(e)>previous P_(e), Flag=1 (new source ON)

3. If current P_(e)<previous P_(e), Flag=2 (source OFF).

The Flag value is transmitted over line 228 to the port allocation logic222, which is called into operation only if Flag=2, indicating that asource has been lost. The function of the port allocation logic 222 isto determine which of the active ports 124 was last processing signalsfrom the source that has just been lost. The basic principle employed tomake this determination is to identify which port has a steering vectorwith the greatest component in the current noise subspace. Each signalthat contributes to the measurements has a steering vector that isorthogonal to the current noise subspace determined from the samplecovariance matrix {circumflex over (R)}. (In a simple three-dimensionalspace, one could think of a first signal eigenvector aligned with thex-axis direction and a second signal eigenvector aligned with the y-axisdirection. The signal subspace for the two active signals includes thex-axis and y-axis directions. The noise subspace is the space defined byall remaining axes in the space. In this case, the noise subspaceeigenvector is in the z-axis direction.)

When a signal disappears and drops out of consideration, the currentnoise subspace then includes the space previously occupied by thesignal. In the three-dimensional example, if the x-axis signaldisappears, leaving only the y-axis signal, the noise subspace isredefined to include a plane in the x and z directions. To recognizewhich signal was lost, the port allocation unit uses the steering vectorestimates from the previous data block (indicative of the active sourcesbefore one was lost), and projects these vectors into the noise subspaceas defined for the current data block. The steering vector from theprevious data block that lies completely in the current data block noisesubspace, or the one that has the largest component in the noisesubspace, is determined to be the signal that was lost between theprevious and current data blocks. Again, using the three-dimensionalexample, if x-axis signal disappears and the new noise subspace isredefined be the x-z plane, then projection of the previous x and ysignals into the current noise subspace results in a finding that thex-axis signal, lying wholly in the current noise subspace, is the signalthat was lost.

More specifically, the logic 222 obtains the steering vector estimatesfrom all of the ports that were active in the previous data block, andnormalizes them (i.e., scales them have a unit norm). Steering vectorestimates are obtained for only the first (P_(e)+1) ports in order ofdecreasing capture strength. The port allocation logic is concerned withthe direction of the steering vectors in space and any differences inmagnitude arising from different signal strengths should be eliminated.Then the normalized steering vectors are projected onto the currentnoise subspace, as provided from the preprocessing computer 120 in theform of noise subspace eigenvectors E_(n). For example, if a_(k)(m−1) isa steering vector estimate from the kth port that was active in theprevious block, then the port allocation logic 222 computes the“leakage” of the steering vector of the kth port into the current noisesubspace from:

||E_(n)E_(n) ^(H)a_(k)(m−1)/||a_(k)(m−1)|| ||.

The logic then declares the port that has the greatest leakage into thenoise space to be inactive by setting its adaptation flag to zero. Alsothe port's capture strength is set to a value MIN, which is a systemparameter set to some very low value, such as 0.001. It will be recalledthat the capture strength is computed in each active port as describedearlier. However, the controller 222 can overwrite the previouslycomputed value when it is determined that a port has become inactive.

For example, suppose that five ports are available for use, with threesources present in the previous data block. Port 1 was locked ontoSource 3, Port 3 was locked onto Source 1, and Port 5 was locked ontoSource 2. Assume further that Source 3 turned off just before thecurrent block and that the following capture strengths were determinedfor the previous and current data blocks:

Previous Previous Current Current Port Source Capture Source Capture No.No. Strength No. Strength 1 3 0.99  — 0.001 2 — 0.001 — 0.001 3 1 0.9951 0.995 4 — 0.001 — 0.001 5 2 0.98  2 0.98 

In the previous data block, Port 2 and Port 4 were inactive and hadtheir capture strengths set to 0.001. When Source 3 turned off, the portallocation logic 222 determined that Port 1 had lost its signal, usingthe analysis discussed above.

Inherent in the list of capture strengths is a priority list of ports(i.e., a list of port numbers in order of decreasing capture strength).Therefore, the priority list based on the previous data block is[3,1,5,2,4] and the priority list based on the current data block is[3,5,1,2,4]. The convention adopted is that, when ports have identicalcapture strengths, they are assigned priorities based on port numbers.The priority list determination logic 224 generates the priority list inthis manner, based on the capture strength list transferred from theport allocation logic 222. The priority list is used by the adaptationflags logic 226 to generate a list or vector of adaptation flags. Theadaptation vector contains L elements, where L is the number of physicalports in the system. In the example given above, the adaptation flagsvector for the previous data block is [1,0,1,0,1] and for the currentdata block is [0,0,1,0,1]. The adaptation flags vector is supplied tomultiple port signal recovery unit (124, 126, FIG. 9), and specificallyto the signal extraction ports 124. The priority list is also suppliedto the multiple port signal recovery unit, and specifically to theorthogonalizer 126, which will be discussed in the next descriptivesection.

The purpose of the priority list is to facilitate an orderly allocationof signal sources to ports, from the lowest port number to the highest.Further, when a signal source turns on, it is desirable that the mostrecently freed port be made available for assignment to the new signal,to provide continuity when a source turns off and on again without achange in the status of other sources.

If there is a new source (i.e., Flag=1), then this unit first obtainsthe previous block steering vector estimates from the ports that areinactive in the previous block (Ports P_(e) to L in the priority list),and normalizes them to have unit norm. It then projects these steeringvectors onto the current noise subspace. For example, if a_(k)(m−1 ) isa steering vector estimate from a port that was inactive in the previousblock, then the Port Allocation Unit computes the port's leakage from(here ||a_(k)(m−1)| denotes the norm of vector a_(k)(m−1)):

 ||E_(n)E_(n) ^(H)a_(k)(m−1)/||a_(k)(m−1)|| ||,

and declares the port which has the minimum leakage as active (sets itsadaptation flag to one) and overwrites the port's capture strength with2×MIN, where MIN is a system parameter that is nominally set to 0.001.This is done to make the newly activated port be the last in thepriority list. (The capture strength of a port is computed by themultiple port signal recovery unit described earlier in thisspecification. The controller, however, can overwrite the computed valueas described above.)

3.5 The Orthogonalizer:

As already briefly discussed, the orthogonalizer 126 functions to ensurethat each port is consistently assigned to process only one signalsource, which is to say that each active port captures a differentsource. The orthogonalizer 126 receives a normalized cumulant vectorfrom each active port, the vector being represented by b_(k) for the kthport. The orthogonalizer 126 outputs back to each port an orthogonalizedcumulant vector, which is d_(k) for the kth port. The orthogonalizeralso receives the priority list from the signal recovery controller 122,so has knowledge of the identities of the active ports and theirrespective associated capture strengths, and also receives the estimatednumber of signals P_(e), from the preprocessing computer 120.

The orthogonalizer forces the active ports to capture different sourcesby orthogonalizing their cumulant vectors, which, in turn, are estimatesof the steering vectors in the dimensionally reduced space. (It will berecalled that, in the preprocessing computer 120, the dimensionality ofthe data is reduced from M, the number of sensor elements, to P_(e), theestimated number of sources.) Ideally, the cumulant vectors for activeports should be orthogonal to each other, to cause the ports to capturedifferent source signals and to prevent two ports from locking up on thesame source signal. From the priority list and the estimated number ofsignals P_(e), the orthogonalizer forms a P_(e) by P_(e) matrix Z fromthe active port steering vectors b_(k) such that the kth column of thematrix Z is the steering vector of the port that is the kth item in thepriority list. The orthogonalizer uses a known procedure known as theclassical Gram-Schmidt (CGS) algorithm to perform the orthogonalizationoperation. The Gram-Schmidt algorithm is described in a number of textson matrix computations, such as Matrix Computations, by Gene H. Goluband Charles F. Van Loan (The Johns Hopkins University Press, 1983), pp.150-154.

As applied to the present system, the Gram-Schmidt algorithm is appliedto the matrix Z to obtain a decomposition of the type:

Z=QR,

where Q is an orthogonal matrix (Q^(H)Q=I), and R is an upper triangularmatrix. Although the Gram-Schmidt orthogonalization procedure is usedbecause of its simplicity, there are alternatives that might also beused in the invention, such as QR-decomposition and the ModifiedGram-Schmidt (MGS) procedure. After the orthogonal matrix Q isdetermined, its columns are shipped back to the ports as orthogonalizedcumulant vectors d_(k). Specifically, the kth column of the Q matrix issent back as d_(k) to the port that is the kth entry in the prioritylist. Regardless of the method used, the effect of the orthogonalizer isto produce a set of P_(e) cumulant vectors that are orthogonal to eachother.

3.6 Operation at an Inactive Port:

When a port is determined to be inactive, as indicated by a zeroadaptation flag, the port performs three simple functions, as shown inFIG. 14. First, its output signal g_(k)(t) is set to zero. Second, itscapture strength c_(k) is set to a minimum value MIN. Finally, the laststeering vector a_(k), estimated in the port just before it becameinactive, is stored in a memory device associated with the port, tofacilitate recapture of the same signal that was lost, if it should turnon again in the near future. More specifically, the adaptation flaginput to the inactive port is delayed by one data block time. Then,using the delayed adaptation flag, the port stores a steering vectoreither from two blocks earlier, if the delayed flag has a value of 1, orone block earlier, if the delayed flag has a value of zero.

4.0 Alternate Embodiment Using Covariance Inversion Cure (CiCURE):

The basic cumulant recovery (CURE) system described in Section 3.0 useseigendecomposition in preprocessing and is referred to as eigenCURE(eCURE) for convenience. Another variant of the CURE system usescovariance inversion instead of eigendecomposition and is referred to ascovariance-inversion CURE (CiCURE). CiCURE is best thought of as alow-cost approximation to the high-performance eCURE system. As such, itshares most of the same advantages over standard CURE as the eCUREmethod.

Certain conditions must be met in order for CiCURE to mimic eCURE andrealize the same advantages. The conditions are:

Sensor noises must be additive Gaussian noise (eCURE assumesindependent, identically distributed, additive Gaussian noise).

Received signal powers must be much greater than the noise.

Sample covariance matrix and its inverse used by CiCURE must be accurateenough to prevent leakage into noise subspace.

Under stationary or steady-state signal conditions, this implies a needfor a sufficiently long processing block size. Under these conditions,the prewhitening transformation used in CiCURE is a good approximationto that used in eCURE, and the two systems have similar performanceproperties. This section of the specification describes the componentsof a signal separation/recovery system that is based on the CiCUREalgorithm.

The CiCURE signal separation/recovery system incorporates a spatialprewhitening transformation based on the inverse of the input samplecovariance matrix (this operation is performed by using aneigendecomposition in the eCURE). The received signal data is filteredor transformed by the prewhitening operation in the CiCURE, unlikeeCURE. The prewhitening done in the CiCURE is implicit in themathematics of the signal recovery ports, and it is only necessary tocompute a matrix decomposition of the input sample covariance matrix.This latter operation is done in a preprocessor, whose output is madeavailable to all of the signal recovery ports. Key characteristics ofthe implementation described below are that the iterative convergence ofthe CURE algorithm is realized over several blocks instead of within asingle block, and “high-priority” ports converge sooner than“low-priority” ports.

A signal recovery system based on the CiCURE method is simpler than onebased on the eCURE. There are two main architectural components to themethod: a preprocessor unit which computes a matrix decomposition and aset of signal recovery ports hierarchically arranged.

In addition, there can be two optional units: demodulators, to completethe recovery of the separated signal for the purpose of recording orlistening, and a direction-finding (DF) search unit to providecopy-aided DF.

A block diagram for an overall signal recovery system based on theCiCURE method is shown in FIG. 15. The details of the preprocessor 120′and the signal recovery ports (124.1, 124.2, etc.) that are unique tothe CiCURE method are described below. All other system details are aspreviously described for the eCURE system.

The preprocessor 20′ computes a matrix decomposition of the input samplecovariance matrix. It does this on a block-by-block basis by firstcomputing the sample covariance matrix of the array snapshots within aprocessing block and then computing the Cholesky decomposition of thesample covariance matrix. The Cholesky decomposition is output to thesignal recovery ports 124.1, 124.2, 124.3, which use this information toadapt their weight vectors to separate the cochannel source signals.

The signal model for the narrowband array case is described by thefollowing equation:

r(t)=As(t)+n(t)

where r(t) denotes the array signals collected by M sensors.

We assume that for each block N snapshots are collected for analysis andthat there are P sources contributing to the measurements. We alsoassume that the measurement noise, n(t), is spatially white and noisepower at each sensor is identical but unknown and it is denoted by σ².The preprocessor 20′ first forms the sample covariance matrix from thesnapshots according to:$\hat{R} = {\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}{{r(t)}{{r^{H}(t)}.}}}}$

After forming the sample covariance matrix, the Cholesky decompositionis performed:

{circumflex over (R)}=LL^(H),

where L is a lower triangular matrix with positive diagonal terms. L issent to the signal recovery ports 124.1, 124.2, 124.3, as indicated inthe drawing.

In the CiCURE structure, there is no controller unit to detect sourceON/OFF transitions as in the eCURE system. The signal recovery portshave a predetermined hierarchy or priority order. The first port hashighest priority and so on. Therefore, CiCURE is not able to compensatefor dynamic changes in the signal environment as can the eCUREalgorithm. Each port receives as input the current block steering vectorestimates from the ports that are higher in priority, the sensor signaldata, and the Cholesky decomposition of the sample covariance matrix forthe current block. Each port outputs the recovered signal and associatedsteering vector for a captured source.

FIG. 16 shows the operations of a single signal recovery port. Supposethe higher-priority ports produce a set of steering vectors for thecurrent block, defined as {a₁(m), . . . , a_(k-1)(m)}. The weight vectorto produce p_(k)(t) for the current block (the mth block) is determinedby modifying the MVDR weights for the kth port on the previous block bya computation carried out in the kth port as described next. The firststep is to compute Gram-Schmidt orthogonalized weights (v_(k)(m))according to:${v_{k}(m)} = {{w_{k}\left( {m - 1} \right)} - {\sum\limits_{l = 1}^{k - 1}{\left( {{a_{l}^{H}(m)}w_{k}} \right){{a_{l}(m)}/{{{a_{l}(m)}}^{2}.}}}}}$

The port then uses the Gram-Schmidt orthogonalized weights (v_(k)(m)) todetermine the waveform p_(k)(t) according to

p_(k)(t)=v_(k) ^(H)(m)r(t).

Next, a vector, a_(k)(m), of sample cross-cumulants involving thiswaveform is computed having components:

[a_(k)(m)]_(l)=cum(p_(k)(t),p_(k) ^(*)(t),p_(k) ^(*)(t),r_(l)(t)),1≦l≦M,

in which, [a_(k)(m)]_(l) is the lth component of a_(k)(m). This vectorprovides an estimate of a source steering vector and is sent to all theports that have lower priority than the kth port and the optionaldirection-finding unit. In addition, the MVDR weight vector for the kthport is determined using a_(k) and L.

The MVDR weight vector is computed in a two-step procedure that exploitsthe lower triangular structure of the Cholesky decomposition. First, thetemporary solution, u_(k) is computed by solving the linear system ofequations:

Lu_(k)(m)=a_(k)(m).

Next, the MVDR weights are computed by solving the second linear system:

L^(H)w_(k)(m)=u_(k)(m)/||u_(k)(m)||.

It is necessary to maintain phase continuity with the weights of theprevious block. This requirement resolves the complex phase ambiguityinherent in the blind signal separation problem, which would otherwisecause “glitches” in the recovered signals at the block boundaries.Therefore, before using the weight vector w_(k)(m) estimate the signalwaveform, the complex phase ambiguity is resolved by computing the scalefactor:

c _(k)(m)=w _(k) ^(H)(m)w _(k)(m−1)/ |w _(k) ^(H)(m)w _(k)(m−1)|,

and then scaling the MVDR weights according to:

w_(k)(m)=c_(k)(m)w_(k)(m).

This operation forces the current and previous block signal extractionweights to have a real inner product (i.e., no abrupt phase rotation atthe block boundary). This method eliminates block-to-block phasediscontinuities, leaving only a bulk phase rotation ambiguity that isconstant over all blocks recovered by the port. This bulk phase rotationis unimportant to the recovery of analog AM and FM modulated signals;however, for digital modulations, its removal is desired. Section Asubsequent section on phase rotation equalization presents a method fordoing so. For now, we skip over this minor detail.

Using the modified MVDR weights, the waveform estimate is computedaccording to:

g_(k)(t)=w_(k) ^(H)r(t)

The recovered waveform is available for subsequent processing, which mayconsist of recording of the predetected waveform, demodulation, signalrecognition, or other operations.

The current block weights are fed into a one-block delay unit whichmakes them available to the Gram-Schmidt orthogonalization unit as theinitial weights for processing the next block. Key characteristics ofthis implementation are that the iterative convergence of the CUREalgorithm is realized over several blocks instead of within a singleblock, and high-priority ports converge sooner than low-priority ports.

5.0 Alternate Embodiment Using Pipelined Cumulant Recovery (pipeCURE):

This section describes a variant or extension of theeigendecomposition-based CURE (eCURE) system, which will be called thepipelined eigenCURE (pipeCURE) system. The eigenCURE (eCURE) algorithmanalyzes measurements on a block by block basis and has dynamiccapabilities to eliminate port switching and port allocation in the caseof transient sources. Received signal data are filtered or transformedby a prewhitening operation before reaching cumulant based signalseparation processing. eCURE (described in Section 3.0) has severaladvantages over the covariance-inversion CURE (CiCURE) (described inSection 4.0), which uses covariance-inversion instead ofeigendecomposition:

It has better signal separation performance (i.e., better crosstalkrejection at port outputs).

It has guaranteed fast convergence, specifically a superexponentialconvergence rate which is mathematically guaranteed.

It has improved port stability which helps minimize random portswitching.

It can operate with a much wider range of input signal strengths.

This last property is particularly useful when trying to recover a weaksignal in the presence of strong interfering signals.

The pipeCURE system “pipelines” the eCURE algorithm in order to have:

Simpler implementation (no feedback between operational blocks).

An ability to iterate more times over one block of data.

An option to use further eigendecompositions to improve results.

5.1 Overview of the pipeCURE Signal Separator:

The pipeCURE signal separator has three main components, which are shownin FIG. 17: a preprocessor unit 120, which is basically the same as inthe eCURE system, a cumulant matrix computer 240, and a multiple portsignal recovery unit 242. In addition, there are two optional units: thedemodulators 128, to complete the recovery of the separated signals forthe purpose of recording and the direction-finding (DF) search unit 172to provide copy-aided directions of arrival on output line 174.

5.2 Preprocessor Unit:

The preprocessor 120 performs a block-by-block analysis of the elementarray snapshots, determining the eigenvectors, number of signal sources,and signal subspace of the received array measurement data. It filtersthe received array data, transforming it from the M-dimensional sensorspace to the P-dimensional signal subspace. In so doing, the steeringvectors of the transformed sources are made orthogonal to each other inthe range of this projection, and the transformed source powers are madeequal (at high signal-to-noise ratios). The details of the preprocessoroperation were described in Section 3.0 in relation to the eCURE system

5.3 Cumulant Matrix Computer:

In this section, we introduce the cumulant matrix computer, a unit thatcomputes the statistics as required by the iterative blind signalseparation processor. The cumulant matrix computer computes a P²×P²(here we assume the number of sources and its estimate are identical)cumulant matrix C is defined as:

C(P·(i−1)+j,P·(k−1)+l)=cum(y _(i) ^(*)(t),y _(j)(t),y _(k)(t),y _(l)^(*)(t)) 1≦i,j,k,l≦P

With finite samples, this matrix can be estimated as:${C\left( {{{P \cdot \left( {i - 1} \right)} + j},{{P \cdot \left( {k - 1} \right)} + l}} \right)} = {{\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}\quad {{y_{i}^{*}(t)}{y_{j}(t)}{y_{k}(t)}{y_{l}^{*}(t)}}}} - {\frac{1}{N^{2}}\quad {\sum\limits_{t_{1} = 1}^{N}\quad {{y_{i}^{*}\left( t_{1} \right)}{y_{j}\left( t_{1} \right)}{\sum\limits_{t_{2} = 1}^{N}\quad {{y_{k}\left( t_{2} \right)}{y_{l}^{*}\left( t_{2} \right)}}}}}} - {\frac{1}{N^{2}}\quad {\sum\limits_{t_{1} = 1}^{N}\quad {{y_{i}^{*}\left( t_{1} \right)}{y_{k}\left( t_{1} \right)}{\sum\limits_{t_{2} = 1}^{N}\quad {{y_{j}\left( t_{2} \right)}y_{l}^{*}\left( t_{2} \right)}}}}} - {\frac{1}{N^{2}}\quad {\sum\limits_{t_{1} = 1}^{N}\quad {{y_{i}^{*}\left( t_{1} \right)}{y_{l}^{*}\left( t_{1} \right)}{\sum\limits_{t_{2} = 1}^{N}\quad {{y_{j}\left( t_{2} \right)}{y_{k}\left( t_{2} \right)}}}}}}}$1 ≤ i, j, k, l ≤ P

in which the signal vector y(t) is defined as:

 y(t)=T^(H)r(t), where TE_(s)(Λ_(s)σ_(n) ²I)^(−½)=US⁻¹

5.4 Multiple Port Signal Recovery Unit:

The multiple port signal recovery unit 242, receives as inputs thepreprocessed array measurement y(t), the cumulant matrix C theeigenstructure (E, Λ) derived from the array measurements in thepreprocessor 120 and the estimated number of sources (P_(e)) generatedin the preprocessor. Using these input signals, the multiple port signalrecovery unit derives recovered signals for output on lines 132 andsteering vectors for output on lines 170, in accordance with thefollowing equations and steps:

(a) Inputs to multiple port signal recovery unit:

Number of sources, P.

Transformation matrix TE_(s)(Λ_(s)−σ_(n) ²I)^(−½)

Preprocessed signals: y(t)=T^(H)r(t)

Eigenstructure of the covariance matrix: {E_(s),Λ_(s),σ_(n) ²}

Initial estimates of the steering vectors for sources stored as thecolumns of Ã.

Cumulant matrix C:

C(P·(i−1)+j,P·(k−1)+l)=cum(y _(i) ^(*)(t),y _(j)(t),y _(k)(t),y _(l)^(*)(t)) 1≦i,j,k,l≦P

(b) Outputs from multiple port signal recovery unit:

Estimated steering vector Ã that will be used in the next block as astarting point (in place of Ã).

Recovered signals for the analysis block, ŝ ;(t).

(c) Processing in the multiple port signal recovery unit:

1. Transformation of Steering Vectors: Project the steering matrixestimate onto the reduced dimensional space by the transformation matrixT:

{tilde over (B)}=T^(H)Ã

2. Cumulant Strength Computation: Normalize the norm of each column of{tilde over (B)} and store the results in {tilde over (V)}, and thencompute the cumulant strength for each signal extracted by the weightsusing the matrix vector multiplication:

{tilde over (v)}_(m)={tilde over (b)}_(m)/||{tilde over (b)}_(m)||,where {tilde over (v)}_(m) ({tilde over (b)}_(m)) is the mth column of{tilde over (V)} ({tilde over (B)}).${O_{m} = {{\left( {{\overset{\sim}{v}}_{m}^{*} \otimes {\overset{\sim}{v}}_{m}} \right)^{H}{C\left( {{\overset{\sim}{v}}_{m}^{*} \otimes {\overset{\sim}{v}}_{m}} \right)}}}},\quad {{{for}\quad 1} \leq m \leq {P.}}$

3. Priority Determination: Reorder the columns of {tilde over (V)} andform the matrix {tilde over (W)}, such that the first column of {tildeover (W)} yields the highest cumulant strength, and the last column of{tilde over (W)} yields the smallest cumulant strength.

Columns of {tilde over (V)} in descending cumulant strength→{tilde over(W)}

4. Capture of the kth source: Starting with the kth column of {tildeover (W)}, proceed with the double power method followed by Gram-Schmidtorthogonalization with respect to higher priority columns of {tilde over(W)}, i.e., for column k:${{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)} = {{\alpha_{b + 1} \cdot \left( {{{\overset{\sim}{w}}_{k}^{*}(b)} \otimes I_{P}} \right)^{H}}{C\left( {{{\overset{\sim}{w}}_{k}^{*}(b)} \otimes {{\overset{\sim}{w}}_{k}(b)}} \right)}}},{b\quad {is}\quad {the}\quad {iteration}\quad {number}},$

where the constant a_(b+1) is chosen so that the norm of {tilde over(w)}_(k) (b+1) is unity and b is the iteration number. This operation isfollowed by the Gram-Schmidt orthogonalization:${{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)} = {\beta_{b + 1} \cdot \left( {{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)} - {\sum\limits_{l = 1}^{k - 1}{w_{l} \cdot \left( {w_{l}^{H}{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)}} \right)}}} \right)}},{{{since}\quad {w_{l}}} = 1},$

where the constant β_(b+1) is chosen so that the norm of {tilde over(w)}_(k)(b+1) is unity. In the last expression, w_(l) denotes the finalweight vector with source of priority 1.

5. Capture of the remaining sources: Repeat step 4, for each column fora predetermined K times. After iterations are complete for the kthcolumn, declare the resultant vector as w_(k), and proceed with theremaining columns. After all sources are separated, form the matrix Wthat consists of w_(k)'s as its columns.

Converged weight vectors w_(k)'s→form the columns of W.

6. Port Association: After all the power method and Gram-Schmidtiterations are complete, we compare the angle between columns of W andthe columns of {tilde over (V)}.

Calculate the absolute values of the elements of the matrixZ=W^(H){tilde over (V)}.

Take the arccosine of each component of Z.

To find the port number assigned to the first column of W in theprevious block, simply take the index of the largest element of thefirst row of the matrix Z.

For the second column of W, we proceed the same way except this time wedo not consider previously selected port for the first column. Usingthis rule, we reorder the columns of W, such that there is no portswitching involved.

Reorder the columns of W, based on Z→results in {circumflex over (V)}.

7. Steering Vector Adjustment: Due to the blindness of the problem,estimated steering vectors are subject to arbitrary gain and phaseambiguities. The gain ambiguities are corrected by the unit amplitudeconstraint on the columns of {circumflex over (V)}. However, this doesnot prevent phase modulations on the columns of this matrix. To maintainthis continuity, we compute the inner product of each column of{circumflex over (V)} with the corresponding column in {tilde over (V)}and use the resulting scalar to undo the phase modulation, i.e.,

ε_(m)=angle({tilde over (v)} _(m) ^(H) v _(m)), and v _(m) =v_(m)·exp(−jε _(m)).

8. Backprojection: In order to use the current steering vector estimatesfor the next processing block, we need to backproject the steeringvector estimates for the reduced dimensional space to the measurementspace. This yields the estimate of the steering matrix and can beaccomplished as:

Â ;=E_(s)(Λ_(s)−{circumflex over (σ)}²I_(P) _(e) )^(+½){circumflex over(V)}.

Â ; will be used in the next block as Ã as an estimate of the steeringmatrix in the first step of the multiple port signal extraction unit.

9. Beamforming: It is important to note that beamforming for P sourcesrequires a matrix multiplication of two matrices: the {circumflex over(V)} matrix that is P by P, and the reduced dimensional observationmatrix y(t), which is P by N, where N is the number of snapshots.Usually N is larger than P and this matrix multiplication may take along time because of its size. Therefore, it may be appropriate to dofinal beamforming in another processor since it does not introduce anyfeedback. Final beamforming is accomplished as:

ŝ ;(t)={circumflex over (V)}^(H)y(t)

The estimated signals will be sent to the correct post processing unitsbecause of the orderings involved.

6.0 Steering Vector Tracking Method For Situations Having RelativeMotion:

For situations in which there is relative motion of the source,receiving array, or multipath reflectors, it is desirable to generalizethe CURE algorithms to exploit or compensate for the motion. One rathercomplicated way of doing this is to use an extended Kalman filter totrack the changes in the generalized steering vectors derived by theCURE algorithms. Here we present a simpler method which merely involvesusing a variant of the iterative update equation used in the CUREalgorithms. We present two such variants of the update equations, calledα-βCURE and μCURE, that can be used principally to provide an improvedinitial weight vector for each block of array samples (snapshots). Theseupdate equations can be used with any of the CURE algorithms discussedin previous sections (CiCURE, eCURE, pipeCURE). Consequently α-βCURE andμCURE are not independent stand-alone algorithms, but rather areenhancements to CiCURE, eCURE, and pipeCURE that provide greaterstability and less port-switching in dynamic situations.

The iterative update equations are given by:

w _(k+1) =αw _(k)+βvect[cum( . . . )] (α-βCURE) w _(k+1)=(1-μ)w_(k)+vect[cum( . . . )] (μCURE) =w _(k)+μ[vect[cum( . . . )]−w _(k)].

These iterative update equations may be compared to the standarditerative update equation presented in previous sections for CiCURE,eCURE, and pipeCURE.

w_(k+1)=vect[cum( . . . )]. (CiCURE, eCURE, pipeCURE).

In these equations, k is a time index, and w_(k) is the linear combinerweight vector that converges to the generalized steering vector of oneof the input source signals. The iteration on the index k is onindividual snapshots or block of snapshots occurring through time, asopposed to multiple iterations within a block. The α-β and μ updatemethod does not preclude iteration within a block. Indeed, within-blockiteration can be used in conjunction with block-to-block updating orinitialization. Generally, there is no advantage to using the α-βor μupdate equations for within-block iteration. Within-block iterationshould be done by the standard update equation. The α-β and μ updateequations are best used for block-to-block updating, that is, toinitialize a block's weight vector based on the final converged weightvector from the previous block.

Two equivalent forms are given for the μCURE update equation. As isdiscussed below, the first form is most convenient when the purpose isto predict ahead, whereas the second form is most convenient when thepurpose is to average previous data with new data. Although α-βCURE andμCURE appear to involve different update equations, the algorithms areequivalent provided there is a renormalization of the weight vector atevery iteration. Because a weight vector renormalization is always usedin the iterative steps to prevent the weight vector from shrinkingmonotonically, α-β CURE and μCURE are equivalent.

α-βCURE and μCURE updating can be used for determining the initialEGSV(s) of a block of samples subject to eCURE or pipeCURE processing.When used with eCURE, the block initialization can be performed ineither the M-dimensional sensor space or the P-dimensional signalsubspace. FIG. 19 and FIG. 20 show these two cases, respectivly. Theapplication of α-βCURE and μCURE to CiCURE is similar to FIG. 19 but isnot shown.

FIG. 19 shows the relations among various vectors in one cycle of theμCURE update operating in the M dimensions of the sensor space. Fivevectors are shown. w_(k) is the current weight vector at time k and isalso the current estimate of the generalized steering vector a_(k). Thegeneralized steering vector at time k+1 is denoted a_(k+1). a_(k+)1 isthe generalized steering vector of which w_(k+)1 is an approximation.The cumulant vector cum=vect[cum( . . . )] is represented as a vectoremanating from the origin. The scaled difference vector μ[cum - w_(k)]is shown as a vector originating at the tip of w_(k) and extendingthrough the tip of cum. The tip of this vector defines w_(k+)1.

For the case shown, μ is greater than unity, and the algorithm isanticipatory. This form of the μCURE update is useful when the purposeis to predict a generalized steering vector that is varying with time.It is instructive to think of the μCURE update as an equation of motion:

New Position=Old Position+Velocity×Elapsed Time,

where New Position is identified with w_(k+1); Old Position isidentified with w_(k); Velocity is identified with [cum - w_(k); andElapsed Time is identified with μ.

Conversely, if μ is less than unity, the μCURE update functions as anaverager rather than a predictor, putting weight (1-μ) on the currentweight vector w_(k) and weight μ on cum which is an estimate of thegeneralized steering vector a_(k+)1 (or its projection b_(k+)1). In thiscase, the tip of cum would lie to the right of w_(k+)1, and the scaleddifference vector μ[cum - w_(k)] would point to, but not pass through,cum.

FIG. 20 similarly shows the vector relations for one cycle of the μCUREupdate when the iteration is performed in the P dimensions of the signalsubspace. The vectors are similar to those in FIG. 19 with the keydifference being that b_(k) is the projection of the generalizedsteering vector a_(k) into the signal subspace, and w_(k+)1 approximatesb_(k+1), the projection of a_(k+)1. At the conclusion of the iterations,the estimated generalized steering vector may be obtained bybackprojecting the terminal weight vector w into the M dimensionalsensor space.

α-β and μ iterative updating has an advantage in the situation wheresignal sources are persistent but moving (i.e., non-static geometry).CiCURE, eCURE, and pipeCURE are formulated in the batch-processing mode(i.e., array snapshots are processed one block of samples at a time)assuming the source geometry is static during a block. With thesealgorithms, only small changes in source geometry are allowed to occurfrom one block to the next. α-β CURE and μCURE accommodate greaterchanges from block to block by providing for tracking of the generalizedsteering vectors via the α-β tracking method, which is well known in thesonar and radar engineering literature.

In summary, α-βCURE and μ iterative updating have tracking capabilityinherently built in which can be used to improve the performance ofCiCURE, eCURE, and pipeCURE in situations in which EGSVs are changingdynamically. This tracking capability enables the adaptation togeometrical changes that occur gradually over time. Abrupt changes, likethe appearance of new signals or disappearance of old signals, andattendant port switching are a different problem. Detection logic isstill required to mitigate port switching caused by abrupt changes.

7.0 Alternate Embodiment Using Direct or Analytic Computation:

Unlike the iterative methods presented in previous sections, thissection presents a method for separating signals that is non-iterative.It is, in fact, a closed form, analytic solution for computing thecumulant vectors and generalized steering vectors without the need foriteration. Because the method is non-iterative, the issues ofconvergence and convergence rate are no longer of concern. Convergenceis both assured and instantaneous.

In the direct method, the generalized steering vectors for a smallnumber of sources (two in this example) are computed directly as setforth below, using one of two computational methods:

Steps of Operation for Method 1:

Compute the covariance matrix R for M channel measurements:$R = {\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}{{r(t)}{r^{H}(t)}}}}$

Compute the eigendecomposition for R:

R=E_(s)Λ_(s)E_(s) ^(H)+σ_(n) ²E_(n)E_(n) ^(H)

Compute the transformation matrix T:

T=E_(s)(Λ_(s)−σ_(n) ²I)^(−½)

Preprocess the measurements by the transformation matrix:

y(t)=T^(H)r(t)

Compute the four by four cumulant matrix C from y(t):

C(P·(i−1)+j,P·(k−1)+l)=cum(y _(i) ^(*)(t),y _(j)(t),y _(k)(t),y _(l)^(*)(t)) 1≦i,j,k,l≦2

From its definition, the C matrix can be decomposed as:$C = \begin{bmatrix}C_{11} & C_{21}^{H} \\C_{21} & C_{22}\end{bmatrix}$

in which the three matrices {C₁₁,C₂₂,C₂₁} are defined (because ofprewhitening and circular symmetry assumption): $\begin{matrix}{C_{11} = {\begin{bmatrix}{{E\left\{ {{y_{1}(t)}}^{4} \right\}} - 2} & {E\left\{ {{{y_{1}(t)}}^{2}{y_{1}(t)}{y_{2}^{*}(t)}} \right\}} \\{E\left\{ {{{y_{1}(t)}}^{2}{y_{2}(t)}{y_{1}^{*}(t)}} \right\}} & {{E\left\{ {{{y_{1}(t)}}^{2}{{y_{2}(t)}}^{2}} \right\}} - 1}\end{bmatrix} = \begin{bmatrix}c_{1} & c_{2} \\c_{2}^{*} & c_{3}\end{bmatrix}}} \\{C_{21} = {\begin{bmatrix}{E\left\{ {{{y_{1}(t)}}^{2}{y_{1}(t)}{y_{2}^{*}(t)}} \right\}} & {E\left\{ \left( {{y_{1}(t)}{y_{2}^{*}(t)}} \right)^{2} \right\}} \\{{E\left\{ {{{y_{1}(t)}}^{2}{{y_{2}(t)}}^{2}} \right\}} - 1} & {E\left\{ {{y_{1}(t)}{y_{2}^{*}(t)}{{y_{2}(t)}}^{2}} \right\}}\end{bmatrix} = \begin{bmatrix}c_{2} & c_{4} \\c_{3} & c_{5}\end{bmatrix}}} \\{C_{22} = {\begin{bmatrix}{{E\left\{ {{{y_{1}(t)}}^{2}{{y_{2}(t)}}^{2}} \right\}} - 1} & {E\left\{ {{{y_{2}(t)}}^{2}{y_{1}(t)}{y_{2}^{*}(t)}} \right\}} \\{E\left\{ {{{y_{2}(t)}}^{2}{y_{2}(t)}{y_{1}^{*}(t)}} \right\}} & {{E\left\{ {{y_{2}(t)}}^{4} \right\}} - 2}\end{bmatrix} = \begin{bmatrix}c_{3} & c_{5} \\c_{5}^{*} & c_{6}\end{bmatrix}}}\end{matrix}$

Construct the fourth-order polynomial in terms of the complex variables{v₁, v₂}. $\frac{v_{1}}{v_{2}} = {\frac{\begin{matrix}{{{v_{1}}^{2}\left( {{c_{1}v_{1}} + {c_{2}v_{2}}} \right)} + {{v_{2}}^{2}\left( {{c_{3}v_{1}} + {c_{5}v_{2}}} \right)} +} \\{{v_{1}^{*}{v_{2}\left( {{c_{2}v_{1}} + {c_{4}v_{2}}} \right)}} + {v_{2}^{*}{v_{1}\left( {{c_{2}^{*}v_{1}} + {c_{3}^{*}v_{2}}} \right)}}}\end{matrix}}{\begin{matrix}{{{v_{1}}^{2}\left( {{c_{2}^{*}v_{1}} + {c_{3}v_{2}}} \right)} + {{v_{2}}^{2}\left( {{c_{5}v_{1}} + {c_{6}v_{2}}} \right)} +} \\{{v_{1}^{*}{v_{2}\left( {{c_{3}v_{1}} + {c_{5}v_{2}}} \right)}} + {v_{2}^{*}{v_{1}\left( {{c_{4}^{*}v_{1}} + {c_{5}^{*}v_{2}}} \right)}}}\end{matrix}}.}$

This requires the computation of the cumulants {c₁, . . . c₆} defined inthe above item from the measurements.

Solve the polynomial for {v₁, v₂}. There is one trivial solution{v₁,=v₂=0}. Also note that if the vector (v₁,v₂)^(T) is a solution, thenthe vector (−v₂ ^(*),v₁)^(T) is also a solution.

Evaluate the resultant cumulant strengths from the solutions to thepolynomial: (v^(*) ⊗ v)^(H)C(v^(*) ⊗ v), v = [v₁, v₂]^(H).

Determine the solution for the polynomial that results in the highestcumulant strengths to estimate sources. Let this be (v₁,v₂)^(T). Then,(−v₂ ^(*),v₁)^(T) is the the second solution.

For each accepted solution v, we can find the corresponding steeringvector in the M dimensional sensor space:

a=E_(s)(Λ_(s)−σ_(n) ²)^(½)v

Once the steering vector estimates are found as in the previous step:

1. Port association, and

2. Waveform continuity, can be implemented as described in Section 5.0(pipeCURE).

Steps of Operation for Method 2

Compute the covariance matrix R for M channel measurements:$R = {\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}{{r(t)}{r^{H}(t)}}}}$

Compute the eigendecomposition for R:

R=E_(s)Λ_(s)E_(s) ^(H)+σ_(n) ²E_(n)E_(n) ^(H)

Compute the transformation matrix T:

 T=E_(s)(Λ_(s)−σ_(n) ²I)^(−½)

Preprocess the measurements by the transformation matrix:

y(t)=T^(H)r(t)

after which the measurements take the form:

y(t)=T^(H) r(t)=b ₁ s ₁(t)/σ₁ ² +b ₂ s ₂(t)/σ₂ ² +e(t)

Because of prewhitening, we have the following result for the steeringvectors for the two sources in the two dimensional space:${b_{1} = \begin{bmatrix}{\cos \quad \theta} \\{\sin \quad \theta \quad ^{j\quad \varphi}}\end{bmatrix}},{b_{2} = \begin{bmatrix}{{- \sin}\quad \theta \quad ^{{- j}\quad \varphi}} \\{\cos \quad \theta}\end{bmatrix}},{{b_{1}} = {{b_{2}} = 1}},{{{b_{2}^{H}b_{1}}} = 0.}$

Construct the three by two matrix F using five of the six cumulants:$F = {\begin{bmatrix}c_{1} & c_{2} \\c_{2}^{*} & c_{3}^{*} \\c_{4}^{*} & c_{5}^{*}\end{bmatrix} = \begin{bmatrix}{{E\left\{ {{y_{1}(t)}}^{4} \right\}} - 2} & {E\left\{ {{{y_{1}(t)}}^{2}{y_{1}(t)}{y_{2}^{*}(t)}} \right\}} \\{E\left\{ {{{y_{1}(t)}}^{2}{y_{2}(t)}{y_{1}^{*}(t)}} \right\}} & {{E\left\{ {{{y_{1}(t)}}^{2}{{y_{2}(t)}}^{2}} \right\}} - 1} \\{E\left\{ \left( {{y_{1}^{*}(t)}{y_{2}(t)}} \right)^{2} \right\}} & {E\left\{ {{y_{1}^{*}(t)}{y_{2}(t)}{{y_{2}(t)}}^{2}} \right\}}\end{bmatrix}}$

which can be decomposed into the following three matrices:$\begin{matrix}\begin{matrix}{F = \quad {\underset{\underset{G}{}}{\begin{bmatrix}1 & 1 \\{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}} & {{{- \frac{\cos \quad \theta}{\sin \quad \theta}}^{j\quad \varphi}}\quad} \\\left( {\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}} \right)^{2} & {\left( {{- \frac{\cos \quad \theta}{\sin \quad \theta}}^{j\quad \varphi}} \right)^{2}\quad}\end{bmatrix}}\quad \underset{\underset{D}{}}{\begin{bmatrix}{{\frac{\gamma_{4,1}}{\left( \sigma_{1}^{2} \right)^{2}} \cdot \cos^{4}}\quad \theta} & {0\quad} \\0 & {{{\frac{\gamma_{4,2}}{\left( \sigma_{2}^{2} \right)^{2}} \cdot \sin^{4}}\quad \theta}\quad}\end{bmatrix}}}} \\{\quad \underset{\underset{H}{}}{\begin{bmatrix}1 & {{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{{- j}\quad \varphi}}\quad} \\1 & {{{- \frac{\cos \quad \theta}{\sin \quad \theta}}^{{- j}\quad \varphi}}\quad}\end{bmatrix}}}\end{matrix} \\{F = {GHD}}\end{matrix}$

Also form the four by four cumulant matrix C:

C(P·(i−1)+j,P·(k−1)+l)=cum(y _(i) ^(*)(t),y _(j)(t),y _(k)(t),y _(l)^(*)(t)) 1≦i,j,k,l≦2

The symmetries involved in this matrix reduces the number of distinctcumulants to six. In addition, five of the six cumulants necessary arealready computed when we formed F.

Compute the Singular Value Decomposition (SVD) of the matrix F:

1. If the rank of F is zero, then source separation is not possible.

2. If the rank of F is one, then the principal eigenvector can be usedto separate sources: assume e, is the principal eigenvector of F, andlet its components are defined as: ${e_{1} = \begin{bmatrix}\alpha_{1} \\a_{2} \\\alpha_{3}\end{bmatrix}},{{{\alpha_{1}}^{2} + {\alpha_{2}}^{2} + {\alpha_{3}}^{2}} = 1}$

Then, we can obtain the estimate of the first source, using:${g_{1}(t)} = {\begin{bmatrix}\alpha_{1} \\a_{2}\end{bmatrix}^{H}{y(t)}}$

and the second source waveform can be estimated using:${g_{2}(t)} = {\begin{bmatrix}{- \alpha_{2}^{*}} \\a_{1}\end{bmatrix}^{H}{y(t)}}$

3. If the rank of F is two, then the null eigenvector can be used toseparate sources together with a solution of a quadratic equation: let xdenote the 3 by 1 vector that is orthogonal to the columns of of F(which can be obtained using SVD or QR decomposition of F):

x^(H)F=0, x=[x₁,x₂,x₃]^(T)

Then, due to the Vandermonde structure of the columns of G, we canobtain the parameters$\left\{ {{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}},{{- \frac{\cos \quad \theta}{\sin \quad \theta}}\quad ^{j\quad \varphi}}} \right\},$

as the roots of the quadratic equation:

x ^(H) z=0, z=[z ₁ ,z ₂ ,z ₃ ] ^(T) →x ₁ ^(*) +x ₂ ^(*) z+x ₃ ^(*) z ²=0

Since we know the roots of the above equation should be$\left\{ {{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}},{{- \frac{\cos \quad \theta}{\sin \quad \theta}}\quad ^{j\quad \varphi}}} \right\},$

and we have the weight vectors to separate the sources as:$\begin{matrix}{{g_{1}(t)} = \begin{bmatrix}1 \\{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}}\end{bmatrix}^{H}} \\{{y(t)} = {{{s_{1}(t)}/\left( {\sigma_{1}^{2}\cos \quad \theta} \right)} + {0 \cdot {s_{2}(t)}} + {\begin{bmatrix}1 \\{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}}\end{bmatrix}^{H}{e(t)}}}} \\{{g_{2}(t)} = \begin{bmatrix}1 \\{{- \frac{\cos \quad \theta}{\sin \quad \theta}}\quad ^{j\quad \varphi}}\end{bmatrix}^{H}} \\{{y(t)} = {{{s_{2}(t)}{^{{- j}\quad \varphi}/\left( {\sigma_{2}^{2}\sin \quad \theta} \right)}} + {0 \cdot {s_{1}(t)}} + {\begin{bmatrix}1 \\{{- \frac{\cos \quad \theta}{\sin \quad \theta}}\quad ^{j\quad \varphi}}\end{bmatrix}^{H}{e(t)}}}}\end{matrix}$

In addition, we can normalize the weights to conform to the structure ofthe problem: ${v_{1} = {\beta_{1}\begin{bmatrix}1 \\{\frac{\sin \quad \theta}{\cos \quad \theta}\quad ^{j\quad \varphi}}\end{bmatrix}}},{v_{2} = {\beta_{2}\begin{bmatrix}1 \\{{- \frac{\cos \quad \theta}{\sin \quad \theta}}\quad ^{j\quad \varphi}}\end{bmatrix}}}$

in which {β₁, β₂} are determined to make ∥v₁||=∥v₂||=1.

After the weights for signal separation are determined, then it ispossible to compute cumulant strengths using:(v^(*) ⊗ v)^(H)C(v^(*) ⊗ v), v = [v₁, v₂]^(H)

is one of the weight vectors.

For each solution represented byv, we can find the correspondingsteering vector in the M dimensional sensor space:

a=E_(s)(Λ_(s)−σ_(n) ²)^(½) v

Once the steering vector estimates are found as in the previous step:

1. Port association, and

2. Waveform continuity can be implemented as described in Section 5.0(pipeCURE).

8.0 Separation Capacity and Performance When Overloaded:

When the number of incident signals exceeds the capacity of the systemto separate signals, one would expect system performance to degrade.Unlike some other cochannel signal separation methods, the presentinvention is able to operate under such overload conditions.

Cochannel signal separation systems are designed to be able to separateand recover signals provided the number of cochannel signals incident onthe array does not exceed a number that defines the separation capacityof the system. In the present invention, the separation capacity isequal to the number of sensors M in the receiving array. Consequently, acochannel signal separation system based upon this method can have nomore than M output ports. Of course, the number of output ports can beless than M. For instance, a system can have P output ports, where P<M.In this case the system could recover each of P signals from among Msignals incident on the array. Each of the P signals is recovered by adifferent set of beamformer weights. Each such set or weight vectordefines a sensor directivity pattern having M-1 nulls.

Next consider cochannel signals received at the receiving arrayconsisting of a mixture of coherent and noncoherent multipathcomponents. A complete description of how CURE algorithms behave in thepresence of multipath is given in Section 9.0. In brief, signals withnoncoherent multipath components are recovered on more than one outputport. Each output port is associated with a single generalized steeringvector. The system of the invention automatically creates groups ofmutually coherent multipath arrivals and a different generalizedsteering vector is formed for each such group. Each group is treated asan independent signal and counts as one signal against the capacity M.The maximum number of such noncoherent groups that can be incident onthe array for which the system can perform separation and recoveryequals the separation capacity. In the present invention, this capacityequals the number of array elements M.

In overload situations, the cumulant optimization or iteration, which isthe basis of the invention, still converges to a generalized steeringvector of a signal or noncoherent multipath group. Consequently, thesystem of the invention determines up to M generalized steering vectorsfrom which beamforming weights are computed and which can form up to M-1generalized nulls. However, because the number of signals is greaterthan capacity, it is not possible to recover a given signal whilesimultaneously rejecting all other signals by means of generalizednulls. In particular, there are P-M excess signals (or G-M noncoherentgroups) that leak through the beamformers for the recovered signals.These excess signals are rejected merely by the sidelobe suppressionassociated with each beamformer's directivity pattern. The signals thatare captured for recovery by the system tend to be the strongestsignals, while the excess signals tend to be among the weakest. Theexcess signals contribute to the noise floor of the output ports.However, their contribution is minimal because of their low relativepower and the sidelobe attenuation. Consequently, the recovered signalsat the output ports generally have low crosstalk levels and highsignal-to-interference-plus-noise ratio (SINR).

FIGS. 21 and 22 illustrate the overload concept diagrammatically. FIG.21 shows a basestation having a four-element antenna array, whichprovides input signals to a CURE system for separating receivedcochannel signals. Because the antenna array has only four elements, thesystem has a capacity of four channels. The figure also shows fiveusers, designated User A, User B, User C, User D and User E, allattempting to transmit signals to the basestation array. FIG. 22 shows adirectivity pattern associated with the antenna array as conditioned bythe CURE system to receive signals from User A. Because the array hasonly four elements, it can present directivity nulls in only threedirections. In the example, the directivity pattern presents a stronglobe in the direction of User A, to receive its signals, and presentsits three available nulls toward User B, User D and User E. User C,which produces the weakest and most distant signal, cannot be completelynulled out. Similar directivity patterns will be generated for receivingsignals from User B, User D and User E. In each case, the weakest signal(from User C) will not be completely nulled out by the directivitypattern. The system continues to operate, however, and is degraded onlyin the sense that weaker sources exceeding the system capacity cannot berecovered and will produce some degree of interference with the signalsthat are recovered.

A general conclusion is that the various embodiments of the inventionare tolerant of separation capacity (or number-of-signals) overloadconditions. In other words, the various embodiments of the invention are“failsoft” with respect to overload beyond the signal separationcapacities, and the ability to separate signals degrades gracefully asthe number of signals is increased above the separation capacity. Thisproperty distinguishes CURE algorithms from DF-beamforming cochannelsignal copy algorithms, such as MUSIC and ESPRIT , which do not functionwhen overloaded. (MUSIC is an acronym for MUltiple Signal Identificationand Classification, and ESPRIT is an acronym for Estimation of SignalParameters via Rotational Invariance Techniques. For more information onthese systems, see the papers cited in the “background” section of thisspecification.

9.0 Performance of the Invention in the Presence of Multipath:

This section discusses the performance of the CURE systems (CiCURE,eCURE, pipeCURE, etc.) when the signal environment includes multipathpropagation. We confine the discussion to the phenomenon known asdiscrete multipath, as opposed to continuous volumetric scattering,which is more complicated to describe. However, the statements belowapply to volumetric scattering under certain conditions.

Multipath propagation occurs when a signal from a source travels by twoor more distinct paths to arrive at a receiving antenna from severaldirections simultaneously. A novel feature of the present invention isits ability to separate cochannel signals in a multipath signalenvironment.

Multipath propagation is caused by the physical processes of reflectionand refraction. A similar effect is caused by repeater jamming, whereina signal is received and retransmitted at high power on the samefrequency. Repeaters are commonly used in radio communication to fill inshadow zones, such as around hills or inside tunnels, where thecommunication signal does not propagate naturally. Cochannel repeatersare also used in electronic warfare (EW) systems to “spoof” a radarsystem by retransmitting a radar signal with a random delay atsufficiently high power to mask the actual radar return. The amount ofdelay is set in order to cause a false distance to be measured by theradar.

Naturally occurring multipath propagation can consist of a small numberof discrete specular reflections, or it can consist of a continuum ofreflections caused by scattering from an extended object. The variousmultipath components arriving at the receiving antenna will generally besomewhat different. Differences among the various multipath componentsof a given source signal are (1) different directions of arrival (DOAs);(2) different time delays due to the different path lengths traveled;and (3) different Doppler shifts on each multipath component due tomotion of the transmit antenna, receive antenna, or reflecting body.

Array-based cochannel signal separation and recovery systemstraditionally have difficulty working in a multipath environment (i.e.,when the one or more of the arriving signals incident on the array comefrom several distinct directions simultaneously). For example, mostmultiple source DF-beamforming signal copy systems generally do not workproperly or well in a multipath environment, and special techniques suchas spatial smoothing and temporal smoothing must be employed to DF onthe individual multipath components of an arriving signal. The resultantsystem and processing complexities make DF-based cochannel signalrecovery systems largely impractical for reception in signalenvironments characterized by significant multipath propagation. Thecochannel signal separation capability of the CURE family of systemsovercomes these limitations.

To understand how the CURE system behaves in the multipath contextrequires understanding the different types of multipath effects and howthe system handles each type. In general, multipath arrivals of a sourcesignal can be classified as either coherent or noncoherent depending onwhether the arrivals' cross-correlation function computed over a finitetime interval is large or small. Thus the designation coherent ornoncoherent is relative to the length of the measurement interval.Coherent multipath components frequently occur in situations where thescatterering bodies are near either the transmit or receive antennas andare geometrically fixed or moving at low velocities. Noncoherentmultipath is caused by path delay differences and Doppler shiftdifferences that are large compared to the measurement window.

The CURE algorithms recover generalized steering vectors as opposed toordinary steering vectors. An ordinary steering vector is the value ofthe array manifold at a single angle corresponding to a source's DOA.However in a multipath environment the received wavefield that the arrayspatially samples is composed of many plane waves for each sourcesignal. Each source, therefore cannot be characterized by a single DOAor steering vector. We consider how the CURE algorithms behave underthree cases:

Coherent multipath components,

Noncoherent multipath components,

Mixtures of coherent and noncoherent multipath components.

9.1 Performance Against Coherent Multipath:

In a coherent multipath signal environment, the CURE system finds asingle steering vector for each independent signal source. However,these steering vectors do not correspond to the ordinary steeringvectors, which could be innumerable in the presence of many multipathscatterers. Rather, the CURE vectors are generalized steering vectorsthat correspond to the sum of all the mutually coherent multipathcomponents of a signal source incident on the array. In the case of afinite number of discrete multipaths, the generalized steering vectordepends on the relative power levels, phases, and ordinary steeringvectors of the multipath components.

The CURE signal recovery process is blind to the array manifold. Oncethe generalized steering vectors for the cochannel sources have beenobtained, it is unnecessary to convert them to source directions ofarrival (DOAs) by using the array manifold, as would generally be donein a system employing direction finding (DF). Instead, the beamformingweight vectors for signal recovery are computed directly from thegeneralized steering vectors. This is done by one of two methods: (1) byprojecting each generalized steering vector into the orthogonalcomplement of the subspace defined by the span of the vectors of theother sources (by matrix transformation using the Moore-Penrosepseudo-inverse matrix); or (2) by using the Capon beamformer, alsocalled the Minimum Variance Distortionless Response (MVDR) beamformer inthe acoustics literature, to determine the recovery weight vectors fromthe generalized steering vectors. These solutions are both well known inthe signal processing engineering literature. (See, for example, pp.73-74 of Hamid Krim and Mats Viberg, “Two Decades of Array SignalProcessing Research,” IEEE Signal Processing Magazine, vol. 13, no. 4,pp. 67-94, July 1996, ISSN 1053-5888, or Norman L. Owsley, “Sonar ArrayProcessing,” Chapter 3 of Array Signal Processing, S. Haykin (ed.),Prentice-Hall, 1985, 445 pp., ISBN 0-13-046482-1.) If the generalizedsteering vectors are determined perfectly, i.e., no estimation error,then the former solution would provide zero crosstalk (or maximumsignal-to-interference ratio) among the recovered signals at thebeamformer output. The latter solution would provide recovered signalshaving maximum signal-to-interference-plus-noise ratio (SINR).

Each output port of the CURE-based system has a beamforming weightvector that is orthogonal or nearly orthogonal to the generalizedsteering vectors of the cochannel signals that are rejected by theoutput port. Each beamforming weight vector has a correspondingdirectivity pattern that assigns a gain and phase to every possibledirection of arrival. These directivity patterns can have up to N−1nulls, where N is the number of array elements. The nulls can be eitherphysical nulls in prescribed directions-of-arrival (DOAs) or, in thecase of coherent multipath, generalized nulls. Generalized nulls are notdirectional nulls but rather are formed when a directivity patternassigns gains and phases in the directions of the coherent multipathcomponents of an interfering signal such that the components sum tozero. Generalized nulls have a major advantage over physical nulls forcombating cochannel interference from coherent multipath because fewerdegrees of freedom (i.e., fewer array elements) are required to causecoherent multipath components to sum to zero than are required todirectionally null each component separately.

FIG. 23 illustrates these concepts for the case of a transmissionreceived over a single-bounce path, designated multipath arrival A, adirect path, designated multipath arrival B, and a two-bounce path,designated multipath arrival C. The three multipath components areindicated as having steering vectors of amplitude and angle combinationsA₁ ∠α, A₂ ∠β, and A₃ ∠γ, respectively. The CURE signal recovery systemof the invention presents a directivity pattern that assigns a gain andphase to very possible direction of arrival. The gains and phasescorresponding to the three multipath components in this example areshown as D₁ ∠φ₁, D₂ ∠φ₂, and D₃ ∠φ₃. The corresponding recovered signalfor the combination of multipath components is derived from signals ofthe form:

y(t)=[A ₁ D ₁ e ^(j(α+φ) ^(₁) )+A ₂ D ₂ e ^(j(β+φ) ^(₂) )+A ₃ D ₃ e^(j(γ+φ) ^(₃) )]s(t).

Because the coherent multipath components of each output port's desiredsignal are optimally phased, weighted, and combined in the recoveryprocess, the CURE method realizes a diversity gain in the presence ofmultipath in addition to eliminating cochannel interference. The amountof the gain depends on the number and strengths of the distinctmultipaths that are combined.

9.2 Performance Against Noncoherent Multipath:

Multipath arrivals of a source signal are not always coherent andcapable of being combined. The coherency requirement for CURE is thatthe multipath components of a signal must have high cross-correlationcomputed over the duration of a processing block or data collectioninterval. Multipath coherency can be destroyed by large path delaydifferences and large Doppler shift. When this happens (i.e., when themultipath arrivals are noncoherent), the CURE algorithms recognize andtreat the arrivals as independent cochannel signals. The steering vectoris estimated for each arrival, and each arrival is separately recoveredand assigned to a different output port. Thus, multiple recoveredversions of the source signal are formed. It is straightforward torecognize a noncoherent multipath situation because the same signal willbe coming out of two or more output ports, each with a slightlydifferent time delay or frequency offset.

FIG. 24 shows the sensor array complex directivity pattern in asituation involving receipt of two coherent multipath components of adesired signal and a non-coherent signal from an interference source. Asindicated in the drawing, the complex directivity pattern includes anull presented toward the interference source, while the two multipathcomponents are received and combined in the same way as discussed abovewith reference to FIG. 23.

9.3 Performance Against Mixtures of Coherent and Noncoherent Multipath:

In the general case of both coherent and noncoherent multipath, the CUREalgorithms automatically partition the multipath arrivals into mutuallycoherent groups, and determine a generalized steering vector for eachgroup. As in the case of noncoherent multipath, multiple recoveredversions of the source signal are formed. The diversity gain isdiminished relative to what would have been achieved had the multipatharrivals all belonged to a single coherent group. However, the loss ofdiversity gain is offset by having multiple replicas of the recoveredsignal appear at the beamformer's outputs. Moreover, a post-recoverycombining gain is possible by adding the signals at the output portsafter correction for delay and Doppler shift. If the recovered signalsare to be demodulated, this post-recovery combining step would precededemodulation.

FIG. 25 shows the complex directivity pattern formed by the beamformerof the CURE system in a situation similar to that discussed above forFIG. 24, except that the desired signal multipath components arenon-coherent instead of coherent. The CURE system treats the threearriving signals (the non-coherent multipath components and theinterfering signal) as being from separate sources. The directivitypattern shown is the one that would be presented for recovery of themultipath component designated Arrival B. Physical nulls are presentedtoward the other arriving signals.

FIG. 26 shows a slightly different situation, in which the receivedsignals include a first interferor (A), a second interferor (B) havingtwo coherent multipath components, and a desired signal. Interferor A isrejected by a physical null in the complex directivity pattern.Interferor B is rejected by a generalized null in the directivitypattern, such that the algebraic sum of the multipath arrivals of thesignal from Interferor B is zero. If the Interferor B signal arrivalsare characterized by gains and phases A₁ ∠α and A₂ ∠β, and thedirectivity pattern at the angles of arrival of these components hasgains and angles D₁ ∠φ₁ and D₂ ∠φ₂, then the necessary condition forrejection of the Interferor B signals by the generalized null is:

A ₁ D ₁ e ^(j(α+φ) ^(₁) )+A ₂ D ₂ e ^(j(β+φ) ^(₂) )=0.

10.0 Recovering Communication Signals In The Presence Of InterferingSignals:

This section describes how the present invention is used separate andrecover signals received in the presence of other interfering signalsemanating either from a local “friendly” source or from a deliberatelyoperated jamming transmitter located nearby. The apparatus of theinvention includes an antenna array and a cumulant recovery (CURE)processing system, which processes signals received through the antennaarray and produces outputs at multiple ports corresponding to themultiple sources from which signals are received at the antenna array.The processing system converges rapidly on estimates of the desiredsignals, without knowledge of the geometry of the antenna array.

Two primary problems in this area are addressed by this aspect of thepresent invention. In one situation, an interfering transmitter may be“friendly,” that is to say operated necessarily near a radio receiver,or even on the same ship or vehicle. Even if the transmitter isoperating on a different frequency, there is often “spectral spatter”into the receiving band. In a related situation, the interferingtransmitter is not “friendly” and is much more powerful than thecommunication signals to be received and recovered.

In the first case, it might be desired to listen and receive whilejamming and transmitting simultaneously. Normally such simultaneoustransmit and receive operations are not possible, but the friendlytransmitter can be selectively turned off to permit reception. Prior tothis invention, however, true simultaneous operation of the interferingtransmitter and the receiver were often impossible.

In the second case, where a strong jamming signal is not under“friendly” control, recovery of the received communication signalrequires the use of a nulling antenna array. In the past, systems forrecovering a communication signal in the presence of jamming requiredknowledge of the antenna array geometry and did not always provide rapidconvergence on the desired signal solution.]

As shown in the drawings, the present invention pertains to systems forrecovering communication signals in the presence of interfering orjamming signals, whether or not on the same frequency. Morespecifically, as shown in FIG. 27, a receiving antenna 280 may belocated on the same vehicle or vessel as a local transmitter 282, andthere may be a high-powered transmitter 284 located on a nearby friendlyvessel operating at the same or a different frequency. A desired signalis received from another transmitter 106, located on land or on anothervessel, but is subject to interference from the high-powered transmitter284 and from the local transmitter 282. In accordance with theinvention, the receiving antenna 280 is coupled to a cumulant recovery(CURE) processing system 290, which rapidly processes the signals fromthe antenna array 280 and generates outputs on multiple ports,effectively separating the signals received from the high-poweredinterfering transmitters and the desired received signal onto separateoutput channels, as indicated for ports #1, #2 and #3.

As shown in FIG. 28, a related situation is one in which reception atthe receiving antenna 280 is interfered with by a high-powered jammingtransmitter 292, perhaps on an enemy vessel. As in the previous case,the CURE processing system 290 separates and recovers the desired weaksignal on one output port (#2), while the jamming signal is isolated andmay be discarded from port #1.

11.0 Diversity Path Multiple Access (DPMA) Communication:

The blind cochannel signal separation capability of the CURE algorithmscan be used to make possible a new communication channel access scheme:Diversity Path Multiple Access (DPMA). This technique enables the designof new communication networks that can accommodate more userssimultaneously in a given bandwidth allocation.

The demand for communication services has grown steadily over the pastthree decades. To a limited extent this demand has been offset bytechnological improvements that have made new bandwidth available athigher frequencies up to the optical frequency band. Such bandwidthimprovements, however, have been unable to keep pace with the growingdemand for communication, and new communication methods becamenecessary. In response, communication system engineers have developednew methods for communication, including networks, control protocols,channel access schemes, and modulation schemes. The principal goal ofthese developments is to enable more users to use and share acommunication resource simultaneously without degrading the quality orcreating mutual interference.

11.1 History and Prior Art of Multiple Access Communication:

Prior to this invention, communication engineers had six channel-accessschemes at their disposal whereby multiple users in a network couldshare an RF communication channel in order to transmit simultaneously,more or less, to a central receiving site (e.g., cell base station orsatellite). Communication engineers would use any of the followingschemes to enable radio communication between multiple users and asingle or multiple base stations. For a wireless network design, acommunication engineer would pick one or more of the following methodsas the basis for the design.

Frequency division multiplexing/multiple access (FDM/FDMA)

Spatial division multiplexing/multiple access (SDM/SDMA)

Time division multiplexing/multiple access (TDM/TDMA)

Code division multiplexing/multiple access (CDM/CDMA)

Frequency hop multiplexing/multiple access (FHM/FHMA)

Angle division multiplexing/multiple access (ADM/ADMA).

Although the terminology in this technology is still evolving, thefollowing distinction is often made. If two transmissions arecooperative, in the sense of being part of a common communicationnetwork, the term “multiple access” is used. If the transmissions areindependent and not part of a network, the term “multiplexing” iscommonly used. The distinction is minor and we shall largely ignore itin this description.

In FDMA, different transmitting users are assigned to differentfrequencies. More precisely each transmitting user is assigned adifferent spectral slice that doesn't overlap with those of other users.FDMA was historically the first multiplexing/multiple access method todevelop. Its origin is traced back to the beginning of radio, and it isthe basis for radio and television broadcast services, whereby anindividual is able to receive and select among the signals transmittedby many stations. In an FDMA network, the transmitting users signals arenot cochannel, and cochannel interference is thereby avoided.

The remaining five channel access schemes enable two or more users to beon the same frequency at the same time (i.e., user transmitted signalscan be cochannel). The schemes mitigate or prevent mutual interferenceby different means.

SDMA is the cellular concept, which originated at Bell Laboratories (TheBell System Technical Journal, special issue on Advanced Mobile PhoneService, vol. 58, no. 1, January 1979). Users are divided geographicallyinto cells, seven of which are indicated at 300 in FIG. 29. Each cell300 has a base station 302, and the base stations are linked togethervia fixed land lines 304 or point to point microwave links. A centralfacility, the mobile telephone switching office (MTSO) 306, controlsnetwork operation and generally serves as a gateway for tying the mobilenetwork to other communication services such as the public switchedtelephone network (PSTN) 308. The set of base stations 302 and the MTSO306 form the “backbone” infrastructure of the mobile network. Each basestation 302 has a finite set of frequencies for sending and receiving,and adjacent cells 300 have different sets of frequencies. Within eachcell, FDMA is employed to prevent cochannel interference. There are onlya finite number of frequency sets available, and base stations 302 thatare separated by some minimum distance use the same frequency sets.Thus, two transmitting users, indicated at 310 that are on the samefrequency are necessarily in different cells some distance apart. Eachusers signal enters the backbone through a different base station. Thegeographic distance between cells prevents cochannel interference.

SHMA prevents cochannel interference by prohibiting intracell frequencyreuse and allowing only intercell frequency reuse. The remaining fourchannel access schemes TDMA, CDMA, FHMA, and ADMA overcome thisrestriction and enable frequency reuse among users within acell-intracell frequency reuse.

In TDMA, all users transmit on the same frequency. Each transmittinguser is assigned a unique time slot in which to transmit. The averagerate of information transmission equals the peak or instantaneous ratetimes a duty factor which is the slot duration divided by the revisitinterval. Although the users are sending at the same time, TDMA preventscochannel interference because the users do not actually transmitsimultaneously.

CDMA is a form of direct sequence spread spectrum in which the varioususers encode their transmissions with orthogonal or nearly orthogonalspreading sequences. All transmitting users use the same frequency. Inorder to receive a particular signal, a receiver must despread thesignal using the same sequence that was used to spread it at thetransmitter. Because of the orthogonality property, thecross-correlation between any two spreading codes is near zero. For thisreason, the user signals after reception and despreading are free ofcochannel interference. CDMA is the basis of the IS-95 communicationstandard.

FHMA is used to apply frequency hop spread spectrum technology tocommunication networks. A set of frequency hopping (FH) radios operatein the same band on the same hop frequencies and transmit to a centralreceiving facility or base station without mutual interference providedthe radios use non-interfering hop sequences. Unlike CDMA, the requiredsequence property is not orthogonality or low cross-correlation, butrather a mathematical relative of the Latin Square. FHMA can be thoughtof as a dynamic form of FDMA in which the frequency assignments changeregularly.

ADMA, which is shown in FIG. 30, uses multi-source direction finding(DF) and beamforming technology to isolate and recover the signals fromthe transmitting cochannel users in a cell. Each base station 302 isequipped with a receiving array connected to an N-channel receiver (notshown), where N is the number of antennas in the receive array. Thereceived signals are processed by a multi-source DF system 312 todetermine the directions or angles of arrival (DOAs) of the signals on agiven frequency. Any multi-source DF algorithm can be used to performthe DF function, such as MUSIC, ESPRIT, or WSF, all of which are wellknown in the signal processing engineering literature. Each user 310 ischaracterized by a single unique DOA. Beamforming weight vectors arethen computed, as indicated in block 314, from the estimated directionsthat enable the cochannel signals to be recovered (separated andcopied). A transformation matrix, whose rows are the beamforming weightvectors, multiplies the array signals, and the product yields therecovered cochannel signals. Each row of the transformation matrix(i.e., each weight vector) consists of complex numbers that steer thearray to one particular signal while putting directional nulls in thedirections of the other cochannel signals. The transmitted user signalscan be recovered free of cochannel interference provided the users areangularly dispersed such that they have distinct bearing angles measuredat the receive array. ADMA is described in a recent patent by Roy andOttersten (Richard. H. Roy, III, and Bjorn Ottersten, Spatial DivisionMultiple Access Wireless Communication Systems, U.S. Pat. No. 5,515,378,May 7, 1996), but the patent specification uses the term SDMA.

11.2 A New Method of Multiple Access Communication:

The present invention achieves a new method for channel access inwireless communications that is distinct from the six basic methodsdescribed above. The new method is termed diversity path multiple access(DPMA). It overcomes three limitations of ADMA.

First, wireless channels (characterized by their angle spread, delayspread, and Doppler spread) are dominated by multipath. The transmittinguser signals arrive at a base station from a multiplicity of directionssimultaneously. Angle spread arises due to multipath from localscatterers and remote scatterers. The local scatterers are near the userand near the base station. Measurements have shown that angle spreadsfor cellular channels generally lie in the range from 2 to 360 degrees.Therefore multipath cannot be ignored, and the idea that a user's signalarrives from a single unique direction is demonstrably not true. TheADMA concept of a single wave arriving from a single directioncharacterized by a pair of angles for each cochannel signal source isvalid in free-space communications, perhaps, but is not valid forwireless communication networks operating in the ultra high frequency(UHF) band in urban, suburban, or rural environments.

Second, most multi-source DF algorithms generally do not work properlyor well in a multipath environment. Although well-known techniques suchas spatial smoothing and temporal smoothing can be used to DF on theindividual multipath components of an arriving signal, the resultantsystem/processing complexities make such approaches impractical.

Third, even when multipath is absent, ADMA requires the transmittingcochannel users DOAs to be distinct. That is, the angular separationbetween users cannot be zero. The users must be separated in angle fromone another by some minimum angle. The assignment of a frequency toseveral cochannel users must take this geometric restriction intoaccount. This limits the utility of ADMA.

The cochannel signal separation capability of the CURE algorithmsovercomes the limitations of ADMA. CURE algorithms recover generalizedsteering vectors as opposed to ordinary steering vectors. An ordinarysteering vector is the value of the array manifold at a single anglecorresponding to a source's DOA. However, in a multipath environment,the wavefield at the receiving array is composed of more than one planewave for each source signal. Consequently, sources cannot becharacterized unique DOAs or steering vectors.

The CURE signal recovery process is blind to the array manifold. TheCURE algorithms find, for each source, a generalized steering vectorthat corresponds to the sum of all the mutually coherent multipathcomponents of a signal source incident on the array. The generalizedsteering vector depends on the relative power levels, phases, andordinary steering vectors of the multipath components. Formally, eachgeneralized steering vector is a complex weighted sum of the arraymanifold steering vectors at the multipath arrival directions. Thecomplex weights account for path length and attenuation differencesamong the multipath arrivals. In some cases, the multipath structureconsists of a continuum rather than a few discrete components. In suchcases the generalized steering vector becomes an integral of the arraymanifold over all directions. The CURE system determines the generalizedsteering vectors directly from the received signals, not from the arraymanifold. Indeed, the various embodiments of the CURE system do not needthe array manifold to perform signal separation and recovery.

Once the generalized steering vectors for the cochannel sources havebeen obtained, it is unnecessary to convert them to source DOAs by usingthe array manifold, as would generally be done in a system employing DFsuch as ADMA. Using CURE, the beamforming weight vectors for signalrecovery are computed directly from the generalized steering vectors.This is done by one of two methods: (1) projecting each generalizedsteering vector into the orthogonal complement of subspace or span ofthe vectors of the other sources; (2) using the minimum variancedistortioniess beamformer (MVDR) equations to determine the recoveryweight vectors from the generalized steering vectors. These solutionsare both well known in the signal processing engineering literature. Ifthe generalized steering vectors are determined perfectly, i.e., noestimation error, then the former solution would provide zero crosstalk(i.e., maximum signal-to-interference ratio) among the recovered signalsat the beamformer output. The latter solution would provide recoveredsignals having maximum signal-to-interference-plus-noise ratio (SINR).

An illustration of how the invention is used in the context of a DPMAcommunication system is provided in FIG. 31, which shows a single cell300, with a basestation 302 and two users 310. One user (A) reaches thebasestation through multipath propagation, while the other has a directpropagation path to the basestation. A CURE processing system 316receives and processes the signals received by the basestation 302. Incommunicating with cochannel user A the system 316 generates abeamformer directivity pattern that presents a physical null toward theother user, but presents a generalized steering vector that results inboth multipath components from cochannel user A being received andcombined.

A feature of the CURE systems is that the omnipresence of multipathenables the recovery (separation and copy) of signals from sources thathave zero angular separation from the point of view of the receivingbase station array. For example, consider two sources that are collinearwith the base station such that one source lies behind the other.Although the direction to both sources is identical and the ordinarysteering vectors for line of sight propagation are identical, themultipath configurations are entirely different. Therefore, thegeneralized steering vectors of the two sources will be entirelydifferent. This facilitates the separation and recovery of the cochannelsource signals in situations where ADMA cannot work.

Because the coherent multipath components are optimally phased,weighted, and combined in the recovery process, the CURE method realizesa diversity gain in the presence of multipath. The amount of the gaindepends on the number of distinct multipaths that are sufficientlycoherent to be able to be combined.

Not all multipath components of a source will be coherent and capable ofbeing combined. The coherency requirement is that the multipathcomponents of a signal must have high cross-correlation computed overthe duration of a processing block or data collection interval.Multipath coherency can be destroyed by large path delay differences andlarge Doppler shift. When the multipath components of a signal sourceare not all mutually coherent, the CURE algorithms automaticallypartition the multipath arrivals into coherent groupings, and determinea generalized steering vector for each group. When this happens,multiple recovered versions of the source signal are formed. Thediversity gain is diminished relative to what would have been achievedhad the multipath arrivals all belonged to a single coherent group.However, the loss of diversity gain is offset by having multiplereplicas of the recovered signal appear at the beamformer s outputs.

By using the multipath combining feature of the CURE algorithms, a newcommunication network channel access method is achieved: DPMA. In DPMA,the communication path that defines the link from a transmitting user toa receiving base station consists of a weighted combination ofmultipaths. The multipath processing capability of the CURE systemsprovide a practical means for implementing a communication networkemploying DPMA. The CURE algorithms determine the complexmultipath-combining weights of a desired signal automatically,dynamically, and in real time, while rejecting the multipaths ofcochannel other-user signals.

It is important to note the difference between DPMA and ADMA. Theorthodox ADMA concept consists of a single wave arriving from a singledirection for each signal source (i.e., a multipath-free environment). Adirection-finding algorithm is employed to estimate the directionparameter associated with each arriving cochannel signal. Under thismodel, two signals are inseparable if their directions of arrival areidentical (i.e., if the sources are collinear with the receive array).DPMA by contrast operates in a signal environment where multipath is akey feature. DPMA, unlike ADMA, is tolerant with regard to angularseparation between sources. Even collinear sources having zero angularseparation at the receiving array are separable because differentmultipath structures cause the sources to have different generalizedsteering vectors which the CURE algorithms exploit.

12.0 Application To Two-Way Mobile Wireless Communication Systems:

The CURE cochannel signal separation technology is applicable tocommunication networks composed of two-way communication links in whichmultiple transmissions occur simultaneously on the same frequency. Thenatural application for CURE is to the receiving end of a communicationlink provided it is practical to have an antenna array at the receivingend. For a two-way communication link, this would mean having areceiving antenna array at both ends of the link. In many situations,however, it is practical to have an array at only one end of acommunication link. For instance, in personal mobile wirelesscommunication networks, it is impractical to have an array built intothe user's portable handheld units. In such situations, it is possible,under certain conditions, to establish and maintain isolation betweendifferent cochannel users by putting arrays at just one end of thecommunication link.

CURE cochannel signal separation technology can be applied to cellularcommunication systems in which the earth's surface is partitioned intolocalized regions called ‘cells,’ as described above with reference toFIG. 29. Examples of cellular personal mobile wireless communicationsystems are the Advanced Mobile Phone System (AMPS) and Global SystemMobile (GSM). (See The Bell System Technical Journal, special issue onAdvanced Mobile Phone Service, vol. 58, no. 1, January 1979.)

The CURE cochannel system has been described to this point as atechnology used at the receiving ends of communication links providedthe receiving ends have multi-element antenna arrays. In the case ofcellular networks, however, economics dictates that arrays be put atbase stations only. There are two reasons for this:

An antenna array is a large and expensive physical asset, best suited toinstallation at fixed base stations where proper maintenance and repairis possible. Mobile units would be larger and more expensive if theywere required to have antenna arrays built in.

A single base station array can serve many users at once. Since eachcell has more mobile units than base stations, total system cost islower if arrays are employed at the base stations only.

The effect of locating arrays at base stations is to lower the initialequipment purchase cost to the subscribers while increasing theinfrastructure cost, which is spread over all subscribers in the form ofmonthly service charge.

Like the application to cellular networks, CURE cochannel signalseparation technology can be applied to satellite-based personalcommunication networks in which a space-based array on a satellite formsspot beams on the surface of the earth that define regions similar tothe cells formed with terrestrial base stations. All communicationwithin a spot beam is between the mobile users and the satellite.Communication that bypasses the satellite, between two users in the samespot beam, is precluded. Examples of satellite-based personal mobilewireless communication systems are Iridium, Odyssey, and Global Star.

There are several reasons and advantages relative to the use of CUREcochannel signal separation technology in personal mobile wirelesscommunication networks.

The capacity of a network to accommodate users can be increased byemploying intra-cell frequency reuse. CURE technology makes thispossible by means of diversity path multiple access (DPMA) on thereverse links or uplinks from mobile user-to-base stations or satelliteand by transmit beamforming on the forward links or downlinks from basestation or satellite to mobile user.

Apart from capacity improvement, CURE provides diversity gain which,when used with suitable power control algorithms, can enable the mobileusers to maintain reliable communication with less average transmittedpower.

CURE provides general interference immunity not only from other users inthe network but from arbitrary radiated interference, whether deliberateor unintentional.

12.1 Transmit Beamforming:

The selectivity that enables several users to simultaneously share aradio-frequency (RF) channel for transmission can be accomplished bybeamforming at the transmitters instead of at the receivers. In the caseof a cellular network, it is possible, by means of transmit beamforming,for forward link transmissions (from base station to mobile user) to besent out with directivity patterns that reach the intended user whilepreventing reception at other cochannel users. Two methods can be usedto accomplish the requisite transmit beamforming function: switchedfixed beams and adaptive beams. The basic principles of both approachesare known in communication engineering. However, proper operationdepends on integration with the receive beamforming function provided bythe CURE system. This integration is described below.

In the switched beam approach, a transmit antenna array and a set offixed pre-formed beams is available for transmission. The beams areformed by applying, signals with appropriate gains and phases to theantennas. The gains and phases can be created either by a passivebeamforming matrix that is inserted into the signal path ahead of theantenna array. The outputs of the beamforming matrices are then summedin power combiners that drive each array antenna. The preferredapproach, shown in FIG.. 32, eliminates the expense of RF hardwarebeamformers in favor of digital signal processing. In this method, asignal to be transmitted to a user is input on one of N multiple lines320 to one of N sets of multipliers 322. Each set of multipliers has asother inputs a transmit beamformer weight vector, which is derived froma transmit/receive beamformer weight vector computer 324. The lattercomputer receives estimated generalized steering vectors on lines 36from the CURE system and generates receive beamformer weights on lines48 (see FIG. 4A), and transmit beamformer weight vectors on lines 326.

Thus, in each set of multipliers 322, a signal to be transmitted to auser is multiplied by a transmit beamformer weight vector, which is anM-dimensional complex weight vector, where M is the number of antennaelements. For each of N users, the outputs of the multipliers 322 aresummed in a plurality M of summers 328. That is to say, each summer 328sums the contributions of multiple user signals associated with aparticular antenna element. The signal to be radiated by the lth antennaof M elements is the sum of N terms, each being a complex weightedversion of the signal to a different user. An M-channeldigital-to-analog converter (DAC) 330 and linear power amplifier (LPA)332 is used to drive each antenna. This latter method does not requireexpensive analog rf beamforming matrices and power combiner hardware,since the multipliers 322 and summers 328 are digital processingcomponents, as indicated by the envelope 334.

In the switched beam approach, the beamforming weight vectors arepre-computed and stored in memory. Each weight vector can be used tocreate a directional beam that puts transmitted energy in a differentdirection. The set of all such weight vectors provides a family ofpencil beams that covers all directions in the cell. Only one such beamis selected for transmission on each of the L forward links. The methodof selection is described below. The method mitigates, but does noteliminate, cochannel interference because the energy of a signalunintentionally radiated to other cochannel users is suppressed to thesidelobe level of the beam, assuming the other users do not fall intothe main lobe of the beam. For cellular systems that use analogfrequency modulation (FM) on the forward links like the AMPS system inthe United States, there is, in addition to sidelobe suppression, thesignal capture effect of FM discriminators that provides additionalsuppression of unwanted cochannel interference.

A beam is chosen for transmission to a particular user by means oflogical rules embodied in a beam selection algorithm. The objective isto prevent energy from reaching the other cochannel users where it wouldinterfere with the intended signals being sent to those users. Considerthe following set of assumptions, reasonable for many wirelesscommunication services that operate at UHF frequencies:

The base station has separate transmit and receive arrays.

The transmit and receive arrays are geometrically similar (i.e., havethe same shape).

The transmit and receive arrays have the same size-frequency products(i.e., the ratio of the transmit-to-receive array sizes equals the ratioof the receive-to-transmit frequencies).

The transmit and receive arrays are mounted on a common vertical mast.

The dominant multipath scatterers are not in the immediate vicinity ofthe arrays, so that the arrays are in the farfield of reradiation fromscatterers, and the elevation or depression angle of arrival isessentially zero at both arrays.

Under these assumptions, the best beam for sending energy to a givenuser is the one whose beamforming vector is most nearly orthogonal tothe generalized steering vectors of the other cochannel users (thegeneralized steering vectors being those derived from reception of thereverse link signals at the base station). Orthogonality between twovectors is strictly defined as an inner product of zero. However, strictorthogonality is not generally possible. Fortunately, it is often goodenough to pick the beam whose weight vector has the smallest innerproduct with the reverse link generalized steering vectors of the othercochannel users. This beam will radiate minimal sidelobe energy to theother users.

The beam selection criterion for using a fixed switched-beam array canbe stated precisely: Choose the beam that maximizes the ratio of theinner product of the beam vector with the generalized steering vector ofthe intended user divided by the sum of the inner products with thegeneralized steering vectors of the unintended cochannel users.

The method just described uses a fully adaptive array for the reverselink receive function, as implemented by the CURE method, together witha switched-beam array for the forward link transmit function. The keyfeature of this approach is that antenna arrays are employed only atbase stations.

A somewhat different approach would be to perform the transmitbeamforming by using the exact generalized steering vectors as arederived by the receive function. This method requires a transmit arraythat is geometrically similar to the receive array as described above(i.e., the transmit and receive arrays have the same shape but arescaled by the ratio of the receive-to-transmit frequencies). Forexample, in the case of the AMPS analog cellular systems, the transmitand receive frequencies are offset by 45 MHz. Because the total systembandwidth is small compared to the operating frequencies, the 45 MHzoffset can be regarded approximately as a 5 percent difference in scale.By using a scaled transmit array, if a generalized steering vectorobtained by CURE on receive is used for transmit, then the same arraydirectivity pattern will result. Thus, lobes and nulls will be placed atthe same angles. Nulls directed to other cochannel users on receive willalso be directed at the same other users on transmit, thereby enablingthe base station to selectively direct a signal at a particular desiredcochannel user.

In a multipath environment, the desired user and other user signals willgenerally be via diversity paths (i.e., the DPMA concept). In this case,the generalized steering vectors derived by CURE on receive cause thereceive array (and hence the transmit array) to have a complexdirectivity pattern for each user that sums the multipath arrivals ofthe desired signal with complex weights (gains and phases) that causesthem to add in phase, while simultaneously summing the multipatharrivals of each other user cochannel signal with complex weights thatcauses these signals to sum to zero. Thus, other users are rejected bygeneralized nulls or orthogonality rather than by physical nulls atspecific angles. The energy transmitted to a particular user will besent in the same direction as the receive multipath components, with thesame phase and gain relationships. Therefore, the signal will reach theintended user with a substantial signal level via the diversity path.Simultaneously, the signal will reach the other users via multiple pathsthat will sum to zero provided the mobile users are using a simpleomnidirectional antenna for both transmit and receive.

FIG. 33 is block diagram of a transmitter for use in one form of theCURE system. Some of the components of the transmitter have already beenintroduced in the discussion of FIG. 32. A transmit weight vectorcomputation or beam selection module 340 generates on lines 326 atransmit weight vector for each user k. the module 340 generates thetransmit weight vector based on either of the two approaches discussedabove An information signal to be transmitted to user k is modulated ina modulator 342 and then multiplied by the transmit weight vector foruser k, in a set of multipliers 328. The portion of the transmitterincluding the modulator 342 and multipliers 322 is referred to as thetransmit beamformer 344. Next the outputs of the multipliers 322 aresummed in a set of M summers 328, each summer receiving as inputs anantenna element contribution associated with each of the users. Thuseach summer 322 has N inputs if there are N users. The summers 328 arecollectively referred to as a signal combiner 346.

The outputs of the summers 328 are then processed in what is referred toas the air interface 348 of the transmitter. The air interface includesa set of complex digital-to-analog converters (DACs), each of whichproduces two outputs, the in-phase and quadrature components of thecomplex signals. These complex signal components are multiplied by acarrier signal in additional pairs of multipliers. More specifically,each complex output pair from a complex DAC 330 is multiplied by signalsproportional to cos ω_(c)t, and sin ω_(c)t, respectively, where ω_(c) isthe angular carrier frequency. The resulting products in each pair arethen added in summers 352 and coupled to one of the linear poweramplifiers 332, and from there the signals are coupled to an antennaelement 110.

13.0 System For Separating And Recovering Multimode Radio Signals:

This section describes a method and apparatus for mitigatingpolarization effects on propagated radio signals. In the case ofdual-polarized radio transmissions, the effects of apolarization-changing propagation medium are avoided by separating thetwo received signals without regard to their polarization states.

This invention relates generally to radio communications and, morespecifically, to problems that arise, due to natural propagationconditions, when multiple cochannel signals of practically the samefrequency are received at approximately the same time. Propagationconditions may cause may cause unwanted polarization mixing of thesignals. Separating and recovering the original signals posesdifficulties in receiver design.

A related problem is multipath propagation caused by reflections fromatmospheric layers, such as the D-layer, E-layer or F-layer. The problemmanifests itself as frequency selective fading or phase distortion thatlimits the communication capability of high-frequency (HF) signals. Asalready discussed above in Section 9.0, the CURE system handlesmultipath components advantageously by combining all coherent signalsarriving over different paths as a result of reflections from buildingsin an urban environment. Multipath propagation effects caused byatmospheric reflections are handled in exactly the same manner.

In some communication systems electromagnetic propagating waves are usedto carry two independent information signals on different polarizationsof the same carrier signal. These polarizations need not be orthogonal,but do need to be linearly independent relative to two orthogonal“basis” polarizations, e.g. vertical and horizontal linear polarizationsor left-hand circular and right-hand circular polarization. Atraditional problem is that the polarization of a transmitted signal ischanged by the propagation medium so that the signal arrives at thereceiving antenna with a different polarization from the one in which itwas transmitted. The polarization change may be due to reflection fromoblique surfaces, refraction, or the phenomenon of Faraday rotation.Conventional receivers separate differently polarized signals becauseeach receiver has knowledge of the expected polarization states. Whenthe polarization of one or both signals is changed during propagation,the conventional receiver is incapable of properly separating the twosignals.

The present invention separates the received signals without regard totheir possibly changed polarization states. If only one signal isreceived with an unknown polarization is received at a dual-polarizedantenna, the invention can extract the signal and determine itspolarization state. If two signals are sent on orthogonal polarizations,the signal polarizations can be random and not orthogonal at thereceiving site, making reception of either signal subject to cochannelinterference from the other signal. CURE processing solves the problemby separating and recovering up to two independent signals arriving atthe receiving array with differently polarizations, provided only thatthe polarizations are linearly independent (i.e., not identical). Thekey advantage of the invention in this application is that it is “blind”to the polarization states of the received signals. Prior knowledge ofthe polarization state is not needed to separate and recover thesignals. In addition, the CURE approach is fast enough to enable therecovery of signals whose polarization is time-varying.

As shown in FIGS. 34A and 34B, one type of communication system makesuse of dual-polarized signals at the same frequency. For example,transmitters 360A and 360B transmit uplink signals A and B to acommunication satellite 362, which retransmits the signals, withdifferent polarization states, to a dual-polarized antenna 364 on theground at a receiving site. However, an atmospheric layer 366 causespolarization mixing of the two signals, which arrive at the receivingantenna with scrambled polarization states. The received signals areprocessed by a CURE processing system 368, which effectively separatesout the signals A and B without regard to their scrambled polarizationstates. Because of the CURE processing system 368 is “blind” to antennaconfiguration, and to the polarization state of the received signals,separation and recovery of signals A and B can be effected even whenboth have their polarization states altered during propagation from thesatellite transmitter.

14.0 Application to Separation of Signals Transmitted Over “Waveguide”:

This section describes a method and apparatus for separating andrecovering signals transmitted onto a “waveguide.” As mentioned earlier,the term “waveguide” as used in this specification is intended toinclude any bounded transmission medium, such as a waveguide operatingat microwave frequencies, an optical fiber operating at, a coaxialcable, or even twisted-pair conductors operating at lower frequencies.Regardless of the waveguide medium, the signals are received at an arrayof sensor probes installed in the waveguide, and are fed to a cumulantrecovery (CURE) system that separates and recovers the original signalswithout regard to how the original propagation modes may have becomescrambled as a result of transmission along the waveguide.

Waveguides and optical fibers are widely used for the transmission ofmultiple independent cochannel signals simultaneously, by using adifferent propagating mode for each signal. However, due to kinks, bendsand surface and refractive irregularities in the waveguide or fiber, aphenomenon called mode conversion occurs, and the propagated energy isconverted from one mode to another during propagation along thewaveguide or optical fiber medium. Over a long distance of propagation,the signals tend to become scrambled across the propagating modes.

A well known approach to conserving bandwidth is to employ differentpropagating modes for different signals of the same frequency, and torely on the different propagating modes to effect separation of thesignals as the receiving end of the waveguide or optical fiber.Unfortunately, however, mode conversion often occurs, especially in longwaveguides or fibers, as a result of kinks, bends, and surface andrefractive irregularities of the propagation medium. The modes becomescrambled and separation at the receiving end becomes difficult. Foroptical systems, these difficulties are somewhat reduced by the use ofexpensive single-mode fiber.

FIG. 35 shows, by way of example, a computer network employing anoptical fiber 370 and having a plurality of computer workstations 372connected to the fiber by couplers 104, each of which couples aworkstation to the fiber using a different propagation mode, but at thesame optical frequency. In accordance with the invention, a plurality ofprobes 376 are also coupled to the fiber 370, providing three outputsignal lines for connection to a cumulant recovery (CURE) processingsystem 378, which generates separated signals A, B and C at its signalrecovery ports.

15.0 Application to Radio Direction Finding:

This section describes a method and apparatus for finding accuratedirections of multiple radio signal sources without the need for a fullycalibrated antenna array. Signals from the antenna array are processedin a cumulant recovery (CURE) processing system to recover the signalsand obtain estimated steering vectors for the multiple sources. Signalsfrom a subarray of antennas that are calibrated are combined with thesteering vector estimates to obtain accurate directional locations forall of the sources. There need be only as few as two calibrated antennaelements in the subarray.

This invention relates generally to direction finding (DF) systems and,more particularly, to DF systems using arrays of radio antennas.Traditional super-resolution direction finding systems require an arrayof N+1 calibrated antennas and receiving channels to resolve N sourcelocations (directions). Maintaining large arrays of antennas incalibration adds to the cost of the system. Moreover, traditional DFsystems do not always converge rapidly on the direction solutions.

FIG. 36 shows a direction finding system in accordance with theinvention, including an array of antennas 380, only two of which arecalibrated, a CURE processing system 382 and a copy-aided directionfinding system 384. Signals are received from multiple sources 386 atdifferent directional locations with respect to the antenna array 380.As described in detail in the foregoing descriptive sections, the CUREprocessing system 382 separates and recovers the signals from thesources 386 and outputs the recovered signals from separate outputports, as indicated at 388. A by-product of the signal recovery processis a set of steering vector estimates for the multiple sources 386.

Assume that the k^(th) port provides the steering vector estimatea_(k)(m) from its analysis of the m^(th) block of data “snapshots,” andthe steering vector from the calibration table for the bearing θ isdenoted as a(θ). When an antenna array is “calibrated,” a calibrationtable is generated, associating every bearing angle with an antennasteering vector. The dimensionality of a(θ) is equal to the number ofcalibrated sensors, which must be greater than or equal to two. Afterthe steering vectors are estimated, a search is done to estimate thedirections of arrival for the sources captured by the ports.

The bearing θ_(k) of the source captured by the port k is estimated bythe maximizer of the DOA spectrum:$\theta_{k} = {\underset{\theta}{argmax}\quad {\frac{{{a_{k}^{H}(m)}{a(\theta)}}}{{{a_{k}(m)}}{{a(\theta)}}}.}}$

Some alternative methods to that just described were given by B. Agee in“The Copy/DF Approach to Signal Specific Emitter Location,” Proc.Twenty-Fifth Asilomar Conference on Signals, Systems, and Computers, pp.994-999, Pacific Grove, Calif., November 1991. Agee concludes that thecopy-aided DF method gives more accurate DOA estimates than othersystems, such as MUSIC (discussed at length in the Background of theInvention section), and that these estimates require less computationthan does MUSIC. An additional advantage is that only two calibratedsensors are adequate for azimuth estimation since the search isperformed using the estimated steering vector for one source instead ofthe signal subspace. In the case of multipath propagation, more sourcescan be resolved by the copy-aided DF approach than by MUSIC when spatialsmoothing is used.

16.0 Application to Extending the Dynamic Range of Receiving Systems:

This section describes a method and apparatus for extending theeffective dynamic range of a radio receiving system by removing theprinciple products of distortion through the use of a cumulant recoveryprocessing system. A received signal of interest is separated from theproducts of distortion, which are independent of the signal of interest.The signal is forwarded for further processing and the products ofdistortion are discarded, resulting in an extended dynamic range.

The dynamic range is a measure of the useful output of a receiver inrelation to noise and other unwanted components. It is limited by theintermodulation and distortion products that result from analog anddigital nonlinearities. Analog nonlinear distortion products or spurscan be generated due to signal overload or saturation of the firststage, mixer noise, and other sources. Digital systems employanalog-to-digital (A/D) converters that produce nonlinear distortion dueto uniform quantization noise, A/D saturation, non-monotonicity of theA/D characteristic, sampler aperture jitter, and other physical effects.Accordingly, there has been an ongoing need for significant improvementin the dynamic range of a receiver system.

FIG. 37 shows a multichannel receiver 390, receiving signals fromsources 392 through an antenna array 393 and coupling the input signalsto a cumulant recovery (CURE) processing system 394. The receivedsignals, after analog processing and analog-to-digital conversion, havea spectrum that includes a number of products of distortion, in the formof nonlinear spurs in the spectrum, as well as lower-level quantizationnoise across the entire spectrum of interest. The effect of CUREprocessing is to separate and recover (or discard) received signals. Inthis case, the CURE processing system provides an output port for thedesired signal, and generates other outputs corresponding to theprincipal products of distortion, which may be discarded. The resultingspectrum after CURE processing exhibits improved effective dynamicrange, and contains only lower intensity spurs, and low level noise.Accordingly, the invention eliminates a number of distortion products inthe receiver output and provides a desired output signal with fewer andlesser products of distortion.

17.0 Application to High Density Recording:

This section describes a method and apparatus for separating andrecovering data recorded on closely spaced tracks on a recording medium.An array of sensors senses recorded data from multiple trackssimultaneously, and a cumulant recovery processing system separates andrecovers the data from each of the multiple tracks, without crosstalk ormutual interference. Use of the invention permits recording of data atmuch higher densities than is conventional, so that more data can bestored on a recording disk without increasing its physical size.

For space efficiency, magnetic recordings use multiple parallel tracksto record information. Both rotating disks and linear tape use paralleltracks. Tracks can be laid down in the recording medium side-by-side onthe surface and on top of one another at different vertical depths. Onplayback, a playback head attempts to sense individual tracks withoutcrosstalk or interference from adjacent tracks, but at sufficiently highrecording densities and small track sizes, crosstalk becomes asignificant problem. Accordingly, designers of such systems areconstantly seeking to improve playback head performance and theprecision with which the playback head can be positioned to readinformation from each track. The present invention provides forincreased recording density without crosstalk or mutual interferencebetween adjacent tracks.

As shown in FIG. 38, which depicts a recording disk 400 by way ofexample, playback or retrieval of recorded information is effected bymeans of a multisensor array 402, which has individual sensor elementsthat can sense recorded information on more than one tracksimultaneously. In general, N sensors will permit separation ofinformation from N adjacent tracks. In the illustrated form of theinvention, there are three sensors in the array 402 and the array spansacross three adjacent recording tracks. The signals from the threesensors are processed as independent cochannel signals by a cumulantrecovery (CURE) processing system 404, which generates outputs on threeports, corresponding to the signals on the three tracks over which thesensor array is positioned. Depending on the design of the system,selection from among the three outputs may be simply a matter ofchoosing the strongest signal, which should correspond to the trackabove which the array is centered, or utilizing the information in allthree tracks, based on identifying data contained on the tracksthemselves. Accordingly, the invention represents a significantimprovement in recording and playback techniques using high-densityrecording media.

18.0 Application to Complex Phase Equalization:

This section describes a method and apparatus for effecting automaticphase rotation equalization of a quadrature amplitude modulated (QAM)signal received from a transmitter. Because QAM signals are subject toan unknown phase rotation during propagation, de-rotation or phaserotation equalization is required before the received signal can be QAMdemodulated. In this invention, received downconverted QAM signals aresubject to processing in a cumulant recovery (CURE) processing system,which recovers the originally transmitted I and Q signals andautomatically provides phase rotation equalization, without knowledge ofthe amount of rotation. Thus the invention provides the correct amountof phase compensation automatically, even as channel propagationconditions change.

This invention relates generally to communication systems and, moreparticularly, to phase rotation equalization in communication systems.Many communication systems use a form of modulation referred to asquadrature amplitude modulation (QAM) for transmitting digital data. InQAM, the instantaneous phase and amplitude of a carrier signalrepresents a selected data state. For example, 16-ary QAM has sixteendistinct phase-amplitude combinations, which may be represented in a“signal constellation” diagram as sixteen points arranged on a squarematrix. A special case of QAM signals is the phase-shift keyed (PSK)signals for which the instantaneous phase alone represents a selectedstate. For example, 16-PSK has sixteen distinct phase selections, andcan be represented as sixteen equally spaced points on the unit circle.

Transmission of the modulated signal causes an unknown phase rotation ofthe signals, and phase rotation correction, or equalization, is requiredat the receiver before the QAM signals can be demodulated. The presentinvention provides a convenient and automatic approach to effecting thisphase rotation equalization. FIG. 39 shows a conventional transmitter,including a QAM modulator 410 and a transmitter 412. At the point oftransmission, the signal constellation diagram is as shown at 414, withsixteen phase-modulus points arranged in a square matrix. Each point onthe diagram represents a unique data state. At the receiver site, areceiver and downconverter 416 generates I and Q signal components. Thesignal constellation diagram corresponding to these signals is as shownat 418. The constellation has been rotated and must be corrected beforeQAM demodulation can take place. The receiver site also has a cumulantrecovery (CURE) processing system 420 installed between thereceiver/downconverter 416 and a QAM demodulator 422. As will be furtherexplained, CURE processing has the effect of compensating for the phaserotation induced during propagation of the signal to the receiver site,as indicated by the phase- corrected QAM signal constellation at 424.

The in-phase and quadrature components (I and Q) of a digitalcommunication signal are independent and identically distributed at thetransmitter output, as indicated in FIG. 40Aa). If x(t) is the originalcommunication signal (in analytic representation) with in-phasecomponent x_(p)(t) and quadrature component x_(q)(t), respectively, thenx_(p)(t) and x_(q)(t) are statistically independent. After transmission,the channel distorts the transmitted signal and the receiver recovers itwith gain and phase ambiguity (ignoring measurement noise), i.e. if y(t)is the output of the receiver, we have:

y(t)=G exp(jθ)·x(t).

If y_(p)(t) and y_(q)(t) denote the in-phase and quadrature componentsof y(t) respectively, then we can write: $\begin{bmatrix}{y_{p}(t)} \\{y_{q}(t)}\end{bmatrix} = {G \cdot {{\begin{bmatrix}{\cos (\theta)} & {- {\sin (\theta)}} \\{\sin (\theta)} & {\cos (\theta)}\end{bmatrix}\begin{bmatrix}{x_{p}(t)} \\{x_{q}(t)}\end{bmatrix}}.}}$

The gain term G is real-valued and affects only the scale of the signalconstellation, but not the constellation's shape or alignment with the Iand Q axes. Therefore, without loss of generality, we may assume thatG=1.

The effect of transmission is to rotate the entire signal constellationby the unknown phase angle θ, as is shown in FIG. 40B. In order todemodulate the signal correctly, the constellation must be “de-rotated”back to its original position prior to demodulation. This de-rotationoperation must be accomplished by complex phase equalization or phasecorrection of the I and Q signals, which compensates for the distortionintroduced by the communication channel and the lack of phase referencein the receiver downconverter's local oscillator.

The CURE method can be applied to provide a unique solution to theproblem of complex phase rotation equalization. The signal's centerfrequency for downconversion must be known accurately enough so rotationdoes not occur during a processing block. The following two paragraphshelp explain how the CURE system effects complex phase correction:

1) y_(p)(t) and y_(q)(t) are not independent but are uncorrelated. Theabsence of statistical independence is evident by inspection of FIG.40B. Uncorrelatedness is implied by the equation. Because the signalsare uncorrelated, second-order statistics such as cross-correlationfunctions provide no information to correct for the rotation of thesignal constellation.

2) The rotation correction problem can be considered as a blind signalseparation problem in which there are two sensor signals y_(p)(t) andy_(q)(t), each of which is a linear combination of two independentsource signals xp(t) and xq(t). This is precisely a problem model towhich the CURE processing system can be applied. By applying the CUREmethod to the components of the analytic signal, the vector channel isphase equalized, the original independent I and Q signals are recovered,and the received signal constellation is de-rotated. In addition, sincey_(p)(t) and y_(q)(t) are uncorrelated, the covariance matrix used inthe CURE system will be a scaled identity matrix, which simplifies thepreprocessing required in the CURE system.

The principal advantage of using the CURE system as an equalizationtechnique is that the rotation angle θ need not be known. The CUREsystem compensates for the angle automatically to provide independentoutput signals. Moreover, the equalization process adapts as channelconditions change.

The CURE system can be used to adjust the complex phase forsingle-dimensional constellations, such as pulse-amplitude modulation(PAM). In this case, the scenario can be considered as a two-sensor,single-source, signal enhancement problem, which can be handled by theCURE processing system.

9.0 Extension to Wideband Signal Separation:

The present invention is fundamentally a method for separating andrecovering narrowband cochannel signals illuminating a sensorarray.However, it is possible to extend the method to the separation andrecovery of wideband signals. This is accomplished by:

1) Partitioning the wideband spectrum into multiple narrowband segments.

2) Using an array of cochannel processors to perform signal separationin each narrowband segment.

3) Combining the narrowband results to recover the original widebandsource waveforms.

Steps 1 and 2 are straightforward. Step 3, however, is intricate andrequires a special cumulant test to associate the ports at one frequencysegment with the ports at adjacent frequency segments. Methods toaccomplish these steps are described below. The overall method iscapable of separating and reconstructing wideband signals with nospecial constraints on the signals or their spectra other than that thecomponents of the signals in each narrowband segment must benon-Gaussian. A key advantage of this method is that the signal spectraare not required to be gap-free (i.e., have a convex support set).

19.1 Partitioning Wideband Measurements to Narrowbands:

To apply the CURE algorithm to separate the signals, it is necessary todecompose the sensor measurements into narrowband components. Thisdecomposition step is depicted in block 430 of FIG. 41. If r(t) is thearray snapshot at time t, then let r (tf) denote the measurementcomponent filtered by a bandpass filter centered at frequency f. Thebandpass filters are designed to satisfy the expression:${r(t)} = {\sum\limits_{f = f_{l}}^{f_{u}}\quad {{r\left( {t,f} \right)}.}}$

where f₁<f<f_(u) is the wideband analysis spectrun. With this analysisapproach, the signal model for each band is expressed as

r(t, f)=A(f)s(t,f)+n(t,f).

The following important fact about the cross-cumulants of bandpassfiltered signals will be exploited in order to associate the ports ofdifferent signal separation processors operating in different frequencybands:

cum(s_(k)(t,f₁), s_(k)*(t,f₁), s₁(t,f₂), s_(t)*(t,f₂))=γ_(4,k)(f₁,f₂)δ_(k,l),

where γ_(4,k)(f₁, f₂) ≠0 in general, and δ_(k,l) is the Kronecker deltafunction $\delta_{k,l} = \left\{ \begin{matrix}1 & {{{if}\quad k} = l} \\0 & {otherwise}\end{matrix} \right.$

In other words, the cross-cumulant is nonzero for different frequencycomponents from the same source and zero for different sources.

This property is in contrast to the cross-correlation between thecomponents of signals at different frequencies which, except for signalsthat exhibit second-order cyclostationarity, is generally given by

E{s_(k)(t,f₁)s_(l)*(t,f₂)}=σ_(k) ²(f₁)δ_(k,l)δ_(f) ₁ ,f ₂ .

Here, we see that the components of a given signal at differentfrequency bands are uncorrelated and the components of different signalsare uncorrelated.

The advantage of the cumulant property noted above is that it provides amethod (described below) for associating the narrowband parts of awideband signal that is broadly applicable to all signal typesregardless of whether a given signal exhibits 2nd-ordercyclostationarity.

19.2 Signal Separation in Narrowbands:

Each narrowband component (r(t,f)) is fed to a different CURE signalseparation processor 432 that separates and recovers the signals thatcomprise the narrowband component.

Let gk(t,J) be the waveform recovered by the k^(th) port of the CUREsubsystem that operates in band f processing r(t,f). We shall show amethod of determining which narrowband port signals are part of a commonwideband signal, and how to combine these port signals in order toreconstruct and recover the wideband signal.

19.3 Combining Narrowbands:

The problem of combining the recovered narrowband signals to formwideband signals is complicated primarily by the fact that the ports fordifferent bands capture different sources. The combing step is indicatedas block 434 in FIG. 41. In general,${{\hat{s}}_{k}(t)} \neq {\sum\limits_{f = f_{l}}^{f_{u}}\quad {{g_{k}\left( {t,f} \right)}.}}$

Consider first the case in which the wideband analysis band is brokeninto two narrow bands f₁, f₂, and two processors, each equipped with Lsignal extraction ports to independently process the bands.

Suppose the first port of the processor operating in band f₁ hascaptured a signal and we wish to find whether some port of a secondprocessor assigned to a different band f₂ captures the same signal. Thisdetermination is made by the following test. Compute the followingquantity for the active ports (indexed by l) of the second processor:${d\left( {f_{1},{1;f_{2}},l} \right)} = {\frac{E\left\{ {{g_{1}\left( {t,f_{1}} \right)}}^{2} \right\} E\left\{ {{g_{l}\left( {t,f_{2}} \right)}}^{2} \right\}}{{{cum}\left( {{g_{1}\left( {t,f_{1}} \right)},{g_{1}^{*}\left( {t,f_{1}} \right)},{g_{l}\left( {t,f_{2}} \right)},{g_{l}^{*}\left( {t,f_{2}} \right)}} \right)}}.}$

We associate a port of the second processor to the first port of thefirst processor if both ports jointly minimize the quantity and if theminimum is below a threshold. The threshold is set to limit the numberof false association decisions on average. For example, if port 3 of thesecond processor provides a minimum below threshold, then the waveformsfrom the two processors would be combined according to

ŝ ; ₁(t)=g ₁(t,f ₁)+g ₃(t,f ₂).

Conversely, if the maximum is below threshold, then we let

ŝ ;₁(t)=g₁(t,f₁).

In the general case, there are J signal separation processors operatingin J bands, each having active signal energy on up to L output ports.The method of band association in the general case is to first computethe “distance” between all pairs of ports by computing thepseudo-metric:${{d\left( {i,{k;j},l} \right)} = \frac{E\left\{ {{g_{k}\left( {t,f_{i}} \right)}}^{2} \right\} E\left\{ {{g_{l}\left( {t,f_{j}} \right)}}^{2} \right\}}{{{cum}\left\{ {{g_{k}\left( {t,f_{i}} \right)},{g_{k}^{*}\left( {t,f_{i}} \right)},{g_{l}\left( {t,f_{j}} \right)},{g_{l}^{*}\left( {t,f_{j}} \right)}} \right\}}}},$

for l≦i, j≦J, and l≦k, l≦L,

where g_(k)(t,f_(i)) denotes the waveform from the k^(th) port of theprocessor in the i^(th) band. d(i,k; j,l) is not a true mathematicalmetric because it does not satisfy two of the three required metricproperties: d(i,k;j,l)≠0 and the triangle inequality are not satisfied.Nevertheless, as a pseudo-metric, it does enable port associations to befound.

The next step is to associate ports two at a time. This is done with aclustering algorithm borrowed from the field of statistical patternrecognition. An agglomerative hierarchical clustering algorithm is used.Standard algorithms for such clustering are described in textbooks(e.g., Richard O. Duda and Peter E. Hart, Pattern Classification andScene Analysis, John Wiley & Sons, 1973, pp. 228-237). This type ofclustering algorithm searches through the inter-port distances to findthe two nearest ports. If the distance is below a threshold, the portsare “merged” or associated and all distances to the two ports that aremerged are replaced by distances to the new “merged” signal. Thisprocess is repeated until only distances greater than the thresholdremain. Constraints are imposed on the clustering algorithm to preventthe merging of same-band ports because, if the ports are from the sameprocessor, they cannot capture the same signal.

The distances between same-band ports are used to compute the thresholdthat controls whether ports are sufficiently close to permit them to bemerged. The threshold is computed by a statistical L-estimator operatingon the same-band distances. The distances are sorted into ascendingorder, and a particular distance is selected based on its rank. Thisdistance is multiplied by a constant to obtain the threshold. Both therank and the constant depend on J and L and are chosen to maintain theprobability of false port association, P_(fpa), below some smallspecified level (e.g., 0.001).

After the ports are logically merged or associated into clusters, eachcluster will correspond to exactly one wideband signal. The final stepis to recover the waveforms of the wideband signals. This is done byadding together the output port signals from all the ports merged orassociated to each cluster.

20.0 Conclusion:

As described in detail above, the present invention provides a cochannelsignal processing and separation that has many facets. Implementation ofthe basic cumulant recovery (CURE) processing engine may take the formof any of the proposed embodiments, including eCURE (as described inSection 3.0). CiCURE (described in Section 4.0) or pipeCURE (describedin Section 5.0), and its capabilities may be further extended usingα-βCURE or μCURE (described in Section 6.0), wideband processing(described in Section 19.0), or direct (non-iterative) computation(described in Section 7.0).

Applications of a selected form of the CURE processing engine arenumerous, and probably not all have been described here. Of primeimportance is the application of CURE processing to communicationsystems (described in Sections 8.0 through 12.0), and in particular theconcept of diversity path multiple access (DPMA, described in Section11.0), which not only permits operation in the presence of multipathpropagation, but also takes advantage of multiple coherent signals toprovide a diversity gain, and uses a generalized steering vectorrepresentative of all the multipath components to generate correspondingtransmit weight vectors ensuring that each user receives intendedtransmissions, even in the presence of multipath effects. Othercommunication system applications include signal recovery in thepresence of strong interfering signals (described in Section 10.0),recovery of multimode signals that have been subject to unwanted modemixing (described in Section 13.0), and recovery of signal from abounded waveguide of any type (described in Section 14.0). Otherapplications include radio direction finding (described in Section15.0), extending the dynamic range of receiver systems (described inSection 16.0), high density recording (described in Section 17.0), andcomplex phase equalization (described in Section 18.0).

It will be appreciated from the foregoing that the present inventionrepresents a significant advance in all of these diverse fields ofapplication. It will also be understood that, although a number ofdifferent embodiments and applications of the invention have beendescribed in detail, various modifications may be made without departingfrom the spirit and scope of the invention. Accordingly, the inventionshould not be limited except as by the appended claims.

21.0 MATHEMATICAL BASIS FOR THE INVENTION

This section presents the mathematical basis for the invention. Inparticular, it provides theoretical support for FIGS. 1 through 7, whichillustrate the concepts of the various embodiments of the invention. Thesection is organized as follows:

21.1 Signal Model

21.2 Problem Statement

21.3 Insufficiency of Second Order Statistics for Blind Separation

21.4 Cumulants: Definitions and Properties

21.4.1 Cumulants of Real Random Variables

21.4.2 Sample Estimates of Cumulants

21.4.3 Extension to Random Processes

21.4.4 Extension to Complex Random Processes

21.4.5 Properties of Cumulants

21.5 Motivations to use Cumulants

21.6 Derivations for the Cumulant Recovery (CURE) System

21.6.1 CURE Algorithm as a Cumulant Strength Maximizer

21.6.2 CURE Algorithm as a Structured Eigenvector Computer

21.6.2.1 Cumulant Matrix

21.6.2.2 Limitations of Previous Cumulant-based Approaches

21.6.2.3 Cumulant-based Iterative Blind Separation

21.6.2.4 Direct versus Beamforming-based Computation of the CumulantIteration

21.6.3 Reference Signal Exploitation Framework for the CURE Blind SignalSeparator

21.6.3.1 Reference Signal Exploitation Framework

21.6.3.2 Relationships with Other Algorithms

21.6.3.2.1 CURE and CMA Relationship

21.6.3.2.2 CURE and SCORE Relationship

21.7 Convergence of the Blind Signal Separator

21.7.1 Superexponential Convergence

21.7.2 Assessment of Convergence

21.7.3 Initialization using a Simple Cumulant-based EigendecompositionMethod

21.8 Implementation Details and Flowchart

21.9 Simulation Results

21.1 Signal Model:

The signal model for the narrowband array case can be described by thefollowing equation:

r(t)=As(t)+n(t),

in which r(t) denotes the array measurements collected by M sensors. Theith row, jth column element of the steering matrix A denotes theresponse of the ith element to the wavefront(s) arriving from the jthsource. The jth column of A is called as the (generalized) steeringvector for the jth source. We assume for each analysis block, Nsnapshots are collected and there are P sources contributing to themeasurements. We also assume that the measurement noise n(t) isspatially white, and the noise powers at each sensor are identical butunknown and are denoted by σ².

In this invention, we propose approaches for both coherent andnoncoherent sources. In the case of coherent sources, the response ofthe array to the source is measured in terms of a generalized steeringvector, which is defined as a weighted sum of steering vectors from allmultipath directions that a source is contributing to the measurements.Mathematically, if a source is illuminating the array from L paths, thegeneralized steering vector a can be represented as:$a = {\sum\limits_{l = 1}^{L}{\alpha_{l}{a\left( \theta_{l} \right)}}}$

in which α₁ and a(θ_(l)) denote the multipath coefficient and thesteering vector corresponding for the lth multipath arrivalrespectively. In the case of a single arrival (incoherent source), L=1.

21.2 Problem Statement:

Blind source separation/steering vector identification problemstatement:

For the signal model: r(t)=As(t)+n(t)

Determine the steering matrix (blind steering vector identification) andestimate the source waveforms, s(t) (blind source separation), usingonly the array measurements, r(t).

The real difficulty with the blind separation/identification problem isthat we are limited to use only the array measurements but not anyinformation about the array responses. This is the main difference ofblind source separation and conventional direction methods such as MUSICand ESPRIT.

With no information on the array response, the only way to find asolution is to use the array measurements alone r(t). However, in thenext section, we show that conventional second-order statistics basedanalysis of the array measurements is not sufficient to solve theseparation problem in general.

21.3 Insufficiency of Second-Order Statistic for Blind Separation:

In this section, we show that cochannel sources can not be separated bysecond-order statistics alone. Additional analysis is necessary todetermine the steering vectors and waveforms of the sources.

Let us consider the array covariance matrix and assume that the sourcesare independent and the steering matrix A represents the effects ofsource correlation and propagation. Under the independence assumption,the source correlation matrix R_(SS), and the array covariance matrixR=E{r(t)r^(H) (t)}, can be factored as follows:

R=E{r(t)r ^(H)(t)}=AR _(SS) A ^(H) +σ ² I=(AR _(SS) ^(½)) (AR_(SS)^(½))^(H)+σ² I

in which R_(SS)=diag(σ₁ ², . . . , σ_(P) ²), Λ_(n)=σ_(n) ²I. It is clearthat for source separation purposes, estimating either A or AR_(SS) ^(½)is identical, because R_(SS) is a diagonal matrix. We have the followingsingular value decomposition for AR_(SS) ^(½), based on the assumptionof the linear independence of steering vectors:

AR_(SS) ^(½)=USV

where U is an M×P matrix and V is a P×P matrix, and they satisfyU^(H)U=I_(p) and V^(H) V=I_(p). The P×P diagonal matrix S denotes thesingular values (all positive). Assuming that we are after thenormalized source waveforms, s_(c)(t), each component having unitvariance:

s _(c)(t)=R _(SS) ^(−½) s(t), r(t)=AR _(SS) ^(½) s _(c)(t)+n(t)

We can determine least-squares estimates of normalized source waveformsusing the SVD:

ŝ ;_(c)(t)=(R_(SS) ^(½)A^(H)A R_(SS) ^(½))⁻¹R_(SS)^(½)A^(H)r(t)=(V_(H)SU^(H)USV)⁻¹V^(H)SU^(H)r(t)

Using U^(H) U=I_(p) and V^(H) V=I_(p), we have

ŝ;_(c)(t)=(V^(H)S²V)⁻¹V^(H)SU^(H)r(t)=V^(H)S⁻²VV^(H)SU^(H)r(t)=V^(H)S⁻¹U^(H)r(t)

Therefore, to estimate the source waveforms, we need the matrix V inaddition to U and S. But observing the array covariance matrix

R=(USV)(USV)^(H)+σ_(n) ² I=US(VV ^(H))SU ^(H) =US ² U ^(H)+σ_(n) ² I

we see that R is insensitive (blind) to the presence of the matrix V.The unitary matrix V cannot be estimated from the covariance matrixwithout ambiguity: one cannot distinguish between the correct AR_(SS)^(½)=USV and ‘a’ version modified as AR_(SS) ^(½)U_(am)=US(VU_(am))(with U_(am) ^(H)U_(am)=I_(p)), since both choices yield the samecovariance matrix. On the other hand, we can recover U and S from theeigendecomposition of R:

R=US ² U ^(H)+σ_(n) ² I=E _(S)Λ_(S) E _(S) ^(H)+σ_(n) ² E _(n) E _(n)^(H)

from which we isolate the signal-only components

AR_(SS)A^(H)=US²U^(H)=E_(S)(Λ_(s)−σ_(n) ²I)E_(S) ^(H)

with the result

U=E_(S) and S=(Λ_(S)−σ_(n) ²I)^(½)

Here we note that source separation with the covariance matrix is onlypossible in the hypothetical case when the steering vectors areorthogonal to each other (a_(k) ^(H)a_(l)=0, if k≠l) and source powersare all different (σ_(k) ²'s are all different). In this case, V=I_(p),so there is no need to estimate it (AR_(SS) ^(½)=USI_(p)). The sourcepowers can be determined as the diagonal elements of S. If some of thesource powers are identical, then the corresponding diagonal elements ofS will be identical, which in turn forces the corresponding eigenvaluesof the array covariance matrix to be equal. As a result, correspondingeigenvectors will not be uniquely determined: any linear combination ofthe eigenvectors with a common eigenvalue will be an eigenvector for thecovariance matrix with the same eigenvalue, so that there will be noguarantee that the eigenvectors are the true steering vectors.

Using U and S, we can perform a part of the signal estimation task anddecrease the observation dimension from M to P without losing anyinformation about the signals:

y(t)=T^(H)r(t), where TE_(S)(Λ_(S)−σ_(n) ²I)^(−½)=US⁻¹

This transformation can be rewritten as

y(t)=T ^(H)(AR _(SS) ^(½) s _(c)(t)+n(t))=S ⁻¹ U ^(H)(USVs _(c)(t)+T^(H) n(t))=Vs _(c)(t)+T ^(H) n(t)

which results in the modified observation model${y(t)} = {{{\sum\limits_{k = 1}^{P}{\underset{\underset{\overset{\Delta}{=}\quad b_{k}}{}}{\left( {v_{k}/\sigma_{k}} \right)}\quad {s_{k}(t)}}} + {T^{H}{n(t)}}} = {\underset{\underset{\overset{\Delta}{=}\quad {u{(t)}}}{}}{{Bs}(t)}\quad + {T^{H}{u(t)}}}}$

in which the vector v_(k) is the kth column of V. Since V is a unitarymatrix, we have the following relationship between the modified steeringvectors (b_(k)s):${b_{k}^{H}b_{j}} = {\frac{1}{\sigma_{k}^{2}}\quad \delta_{k,j}}$

The signal components of the P-vector y(t) are uncorrelated since

E{u(t)u^(H)(t)}=T^(H)AR_(SS)A^(H)US²U^(H)T=(S⁻¹U^(H))US²U^(H)(US⁻¹)=I_(p)

is an identity matrix. However, this does not mean that sources areseparated; they are combined in a such a way that they are onlyuncorrelated (i.e., with the modified steering vectors b_(k)'s which areorthogonal to each other). Second-order statistics of y(t) provide noinformation about b_(k)'s, which highlights the need for additional(statistical) information for source separation.

The transformation defined by T indicates that multiple, stationary,zero-mean, directional Gaussian sources cannot be separated by anymethod, because the covariance matrix contains all the information aboutthem, and this information (as explained above) is insufficient for thesource separation problem. However, if there is only one Gaussiandirectional source in addition to other non-Gaussian sources, and we candetermine the steering vectors of the non-Gaussian ones, then theGaussian source waveform can be recovered by observing the signalsubspace of the covariance matrix and the space spanned by the steeringvectors of the non-Gaussian sources. Specifically, if the only Gaussiansource is the first one, and the steering matrix can be decomposed asA=[a_(l),A_(r)], in which A_(r) is known (or estimated), we can find theweight vector in the signal subspace that has perfect nulls on all thesources except the first one by computing the principal component of

(I_(M)−A_(r)(A_(r) ^(H) A_(r))⁻¹A_(r) ^(H))E_(S).

21.4 Cumulants-Definitions and Properties:

This section gives an introduction to the definitions and properties ofcumulants. Basic information on higher order statistics (the bispectrumin particular) can be found in A. Papoulis, Probability, RandomVariables, and Stochastic Processes, Third Edition, 688 pp.,McGraw-Hill, 1991. For a more comprehensive introduction to the world ofhigher order statistics, the reader should consult the followingpublications (listed in order of increasing difficulty):

1. A. Papoulis, Probability, Random Variables, and Stochastic Processes,3rd Edition, McGraw-Hill, 1991, 688 pp. [Cf. section 11-7, pp. 389-395.]

2. Chrysostomos L. Nikias and Jerry M. Mendel, “Signal Processing withHigher-Order Spectra,” IEEE Signal Processing Magazine, vol. 10, no. 3,pp. 10-37, July 1993.

3. Chrysostomos L. Nikias and Mysore R. Raghuveer, “BispectrumEstimation: A Digital Signal Processing Framework,” Proc. IEEE, vol. 75,no. 7, pp. 869-891, July 1987.

4. Jery M. Mendel, “Tutorial on Higher-Order Statistics (Spectra) inSignal Processing and System Theory: Theoretical Results and SomeApplications,” Proc. IEEE, vol. 79, no. 3, pp. 278-305, March 1991.

5. Chrysostomos L. Nikias and Athina P. Petropulu, “Higher-Order SpectraAnalysis: A Nonlinear Signal Processing Framework,” PTR Prentice Hall,Inc., 1993.

21.3.1 Cumulants of Real Random Variables:

The cumulants of a set of random variables give an alternate descriptionto statistical moments. The underlying idea behind cumulants is simple.The probability density function (PDF) of a sum of independent randomvariables is equal to the convolution of the constituent PDFs. Itfollows that the characteristic function of the sum, which is theFourier transform of the convolution PDF, is the product of theconstituent characteristic functions. Hence, the logarithm of thecharacteristic function is the sum of the constituent log-characteristicfunctions, and the Maclaurin series coefficients of thelog-characteristic function are the sums of the Maclaurin seriescoefficients of the constituent log-characteristic functions. Thesecoefficients are the cumulants. In the multivariate case, jointcumulants are obtained as the generalization of these properties to thecase of random vectors. The multilinearity properties, which aresummarized below in [CP1] to [CP7], are an immediate consequence.

Cumulants can be defined in terms of mathematical expectations ofproducts of random variables (i.e., moments) or terms of partialderivatives of the logarithm of the joint characteristic function of aset of random variables. The definitions that result are equivalent. Ifthe goal is to compute estimates of cumulants, then the definition interms of moments is more convenient. The alternate definition in termsof the characteristic function is more useful for analysis, such as theproof that the nth order cumulants of Gaussian random variables areidentically zero whenever n>2. We present below the basic definitions offirst- through fourth-order cumulants in terms of moments.

Definitions: We give the moment definitions of first- throughfourth-order joint cumulants. Let {w,x,y,z} be zero-mean randomvariables (real or complex) defined on a probability space. Thefirst-order cumulant of a random variable x is its mean. $\begin{matrix}{c_{1}^{x} = {{cum}(x)}} \\{= {E\left\{ x \right\}}}\end{matrix}$

The second-order joint cumulant of two random variables x and y is theircovariance. $\begin{matrix}{c_{2}^{x,y} = {{cum}\left( {x,y} \right)}} \\{= {{E\left\{ {xy} \right\}} - {E\left\{ x \right\} E\left\{ y \right\}}}} \\{{= {E\left\{ {xy} \right\}}},\text{zero-mean~~~~case}}\end{matrix}$

The third- and fourth-order joint cumulants are defined as follows.$\begin{matrix}\begin{matrix}{c_{3}^{x,y,z} = {{cum}\left( {x,y,z} \right)}} \\{= {{E\left\{ {xyz} \right\}} - {E\left\{ x \right\}} - {E\left\{ y \right\}} - {E\left\{ z \right\}}}} \\{{= {E\left\{ {xyz} \right\}}},\text{zero-mean~~~~case}}\end{matrix} \\\begin{matrix}{c_{4}^{w,x,y,z} = \quad {{cum}\left( {w,x,y,z} \right)}} \\{= \quad {{E\left\{ {wxyz} \right\}} - {E\left\{ {wxy} \right\} E\left\{ z \right\}} - {E\left\{ {wxz} \right\} E\left\{ y \right\}} - {E\left\{ {wyz} \right\} E\left\{ x \right\}} -}} \\{\quad {{E\left\{ {xyz} \right\} E\left\{ w \right\}} - {E\left\{ {wx} \right\} E\left\{ {yz} \right\}} - {E\left\{ {wy} \right\} E\left\{ {xz} \right\}} - {E\left\{ {wz} \right\} E\left\{ {xy} \right\}} +}} \\{\quad {{2E\left\{ {wx} \right\} E\left\{ y \right\} E\left\{ z \right\}} + {2E\left\{ {wy} \right\} E\left\{ x \right\} E\left\{ z \right\}} +}} \\{\quad {{2E\left\{ {wz} \right\} E\left\{ x \right\} E\left\{ y \right\}} + {2E\left\{ {xy} \right\} E\left\{ w \right\} E\left\{ z \right\}} +}} \\{\quad {{2E\left\{ {xz} \right\} E\left\{ w \right\} E\left\{ y \right\}} + {2E\left\{ {yz} \right\} E\left\{ w \right\} E\left\{ x \right\}} -}} \\{\quad {6E\left\{ w \right\} E\left\{ x \right\} E\left\{ y \right\} E\left\{ z \right\}}} \\{= \quad {{E\left\{ {wxyz} \right\}} - {E\left\{ {wx} \right\} E\left\{ {yz} \right\}} - {E\left\{ {wy} \right\} E\left\{ {xz} \right\}} -}} \\{\quad {{E\left\{ {wz} \right\} E\left\{ {xy} \right\}},\text{zero-mean~~~~case}}}\end{matrix}\end{matrix}$

Joint cumulants of order no, can be defined as the difference betweenthe corresponding nth-order joint moment of the given random variablesand the nth-order joint moment of n Gaussian random variables that havethe same mean vector and covariance matrix as the given randomvariables.

This definition of a joint cumulant as the difference between thenth-order moment of a random vector and that of a Gaussian random vectorhaving the same mean and covariance matrix is reminiscent of thedefinition of the entropy power of a random process in informationtheory. In that definition, however, the entropy power of a stationaryrandom process is defined as the variance of a Gaussian process havingthe same entropy as the process in question.

21.3.2 Sample Estimates of Cumulants:

Sample estimates of the cumulants are straightforward to define. Let{(w_(i), x_(i), y_(i), z_(i)):i=1,2, . . . n}, be a set of n-tuples(samples) that result from repeated trials of the random experiment thatdefines {w,x,y,z}. Then we can define the sample estimates of thecumulants by $\begin{matrix}{{\hat{c}}_{3}^{x,y,z} = \quad {\frac{1}{n}\quad {\sum\limits_{i = 1}^{n}\quad {x_{i}y_{i}z_{i}}}}} \\{{\hat{c}}_{4}^{w,x,y,z} = \quad {{\frac{1}{n}\quad {\sum\limits_{i = 1}^{n}\quad {w_{i}x_{i}y_{i}z_{i}}}} - {\frac{1}{n^{2}}\quad {\sum\limits_{i = 1}^{n}\quad {w_{i}x_{i}{\sum\limits_{j = 1}^{n}\quad {y_{j}z_{j}}}}}} -}} \\{\quad {{\frac{1}{n^{2}}\quad {\sum\limits_{i = 1}^{n}\quad {w_{i}y_{i}{\sum\limits_{j = 1}^{n}\quad {x_{j}z_{j}}}}}} - {\frac{1}{n^{2}}\quad {\sum\limits_{i = 1}^{n}\quad {w_{i}x_{i}{\sum\limits_{j = 1}^{n}\quad {x_{j}y_{j}}}}}}}}\end{matrix}$

Under mild regularity conditions, the sample estimates of the cumulantsconverge to the cumulants as the number of samples n gets large.

21.3.3 Extension to Random Processes:

We can define the random variables to be the values of a random processx(t) at four instants of time. Or we can define them to be the values offour random processes at the same instant of time, such as the outputproduced by a four-element array whose antennas are simultaneouslysampled to produce an array snapshot. For example, third- andfourth-order cumulant function with arguments t_(k) can be defined by$\begin{matrix}{{c_{3}^{x,y,z}\left( {t_{1},t_{2}} \right)} = {{cum}\left( {{x(t)},{y\left( {t + t_{1}} \right)},{z\left( {t + t_{2}} \right)}} \right)}} \\{= {E\left\{ {{x(t)}{y\left( {t + t_{1}} \right)}{z\left( {t + t_{2}} \right)}} \right\}}} \\{{c_{4}^{w,x,y,z}\left( {t_{1},t_{2},t_{3}} \right)} = {{cum}\left( {{w(t)},{x\left( {t + t_{1}} \right)},{y\left( {t + t_{2}} \right)},{z\left( {t + t_{3}} \right)}} \right)}}\end{matrix}$

21.3.4 Extension to Complex Random Processes:

In narrowband array processing applications, it is often necessary touse complex signals. Therefore, we need a definition for the cumulant ofcomplex random processes. In particular, we will focus on the definitionof fourth-order statistics because the third order statistics of signalsthat are common in array processing applications are zero.

The fourth-order cumulant of four zero-mean, complex random processes{v(t), x(t),y(t),z(t)} is defined in a “balanced” way: $\begin{matrix}{{c_{4}^{w,x,y,z}\left( {t_{1},t_{2},t_{3}} \right)} = \quad {{cum}\left( {{w^{*}(t)},{x\left( {t + t_{1}} \right)},{y^{*}\left( {t + t_{2}} \right)},{z\left( {t + t_{3}} \right)}} \right)}} \\{= \quad {{E\left\{ {{w^{*}(t)}{x\left( {t + t_{1}} \right)}{y^{*}\left( {t + t_{2}} \right)}{z\left( {t + t_{3}} \right)}} \right\}} -}} \\{\quad {{E\left\{ {{w^{*}(t)}{x\left( {t + t_{1}} \right)}} \right\} E\left\{ {{y^{*}\left( {t + t_{2}} \right)}{z\left( {t + t_{3}} \right)}} \right\}} -}} \\{\quad {{E\left\{ {{w^{*}(t)}{z\left( {t + t_{3}} \right)}} \right\} E\left\{ {{y^{*}\left( {t + t_{2}} \right)}{x\left( {t + t_{1}} \right)}} \right\}} -}} \\{\quad {E\left\{ {{w^{*}(t)}{y^{*}\left( {t + t_{2}} \right)}} \right\} E\left\{ {{x\left( {t + t_{1}} \right)}{z\left( {t + t_{3}} \right)}} \right\}}}\end{matrix}$

Furthermore, this cumulant can be estimated from the sample measurementsas: $\begin{matrix}{\begin{matrix}{{{cu}m}\left( {{w^{*}(t)},{x\left( {t + t_{1}} \right)},} \right.} \\\left. {{y^{*}\left( {t + t_{2}} \right)},{z\left( {t + t_{3}} \right)}} \right)\end{matrix} \approx \quad {{\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}{{w^{*}(t)}{x\left( {t + t_{1}} \right)}{y^{*}\left( {t + t_{2}} \right)}{z\left( {t + t_{3}} \right)}}}}\quad -}} \\{\quad {{\frac{1}{N^{2}}\quad {\sum\limits_{t,{v = 1}}^{N}{{w^{*}(t)}{x\left( {t + t_{1}} \right)}{y^{*}\left( {v + t_{2}} \right)}{z\left( {v + t_{3}} \right)}}}}\quad -}} \\{\quad {{\frac{1}{N^{2}}\quad {\sum\limits_{t,{v = 1}}^{N}{{w^{*}(t)}{z\left( {t + t_{3}} \right)}{x\left( {v + t_{1}} \right)}\quad {y^{*}\left( {v + t_{2}} \right)}}}} -}} \\{\quad {\frac{1}{N^{2}}\quad {\sum\limits_{t,{v = 1}}^{N}{{w^{*}(t)}{y^{*}\left( {t + t_{2}} \right)}{x\left( {v + t_{1}} \right)}{z\left( {v + t_{3}} \right)}}}}\quad}\end{matrix}$

21.3.5 Properties of Cumulants:

Seven properties of cumulants, denoted [CP1] through [CP7] are listedbelow. Property [CP1] follows directly from thelog-characteristic-function definition of cumulants. Properties [CP2]through [CP7] are as given in [Jerry M. Mendel, “Tutorial onHigher-Order Statistics (Spectra) in Signal Processing and SystemTheory: Theoretical Results and Some Applications,” Proc. IEEE, vol. 79,no. 3, pp. 278-305, March 1991]. Cumulants can be treated as anoperator, similar to the Riemann integral, the Fourier and Laplacetransforms, and mathematical expectation. This is due to multilinearity,which is a consequence of the following six cumulant properties.

[CP1] Cumulants of jointly Gaussian random variables are identicallyzero, i.e. if{x₁, x₂, . . . , x_(n)} are jointly Gaussian randomvariables, then

cum(x₁,x₂, . . . , x_(n))=0

[CP2] Cumulants of scaled quantities (where the scale factors arenon-random) equal the product of all the scale factors times thecumulant of the unscaled quantities, i.e., if {λ₁, λ₂, . . . , λ_(n)}areconstants, and {x₁,x₂, . . . , x_(n)} are random variables, then${{cum}\left( {{\lambda_{1}x_{1}},\ldots \quad,{\lambda_{n}x_{n}}} \right)} = {\left( {\prod\limits_{i = 1}^{n}\quad \lambda_{1}} \right){{cum}\left( {x_{1},\ldots \quad,x_{n}} \right)}}$

[CP3] Cumulants are symmetric in their arguments, for example

cum(x₁, . . . x_(n))=cum(x_(i) _(l) , . . . x_(i) _(n) )

where {i₁, . . . , i_(n)} is a permutation of {1, . . . , n}. This meansthe arguments of the cumulant can be interchanged without changing thevalue of the cumulant.

[CP4] Cumulants are additive in their arguments, i.e., cumulants of sumsequal sums of cumulants. Hence the name “cumulant.” So, for example,even when x₀ and y₀ are not statistically independent, it is true that

cum(x ₀ +y ₀ , z ₁ , . . . , z _(n))=cum(x ₀ , z ₁ , . . . , z_(n))+cum(y ₀ , z ₁ , . . . , z _(n))

[CP5] Cumulants are blind to additive constants, i.e., if α is aconstant, then

cum(α+z₁, z₂, . . . , z_(n))=cum (z₁, z₂, . . . , z_(n))

[CP 6] Cumulants of a sum of statistically independent quantities equalthe sum of the cumulants of the individual quantities, i.e., if therandom variables {x₁, x₂, . . . , x_(n)} are independent of the random{y₁, y₂, . . . , y_(n)}, then

cum(x₁+y₁, . . . , x_(n)+y_(n))=cum(x₁, . . . , x_(n))+cum(y₁, . . . ,y_(n))

Note that if x_(i) and y_(i) are not independent, then by [CP3], therewould be 2^(n) terms on the right-hand side of cum(x₁+y₁, . . . ,x_(n)+y_(n)). Statistical independence reduces this number to two terms.

[CP7] If some subset of the n random variables {x₁, x₂, . . . , x_(n)}is independent of the rest, then

cum(x₁, . . . , x_(n))=0

We would like to note that these properties apply equally well to thecomplex signals case.

21.5 Motivations to Use Cumulants:

The main motivation to use higher that second-order statistics (orcumulants) is the inadequacy of second-order statistics for the blindsource separation problem. Since the use of second-order statistics isnot sufficient to solve the blind separation problem, it is necessary toexploit other statistics of measurements.

Cumulants have long been used to measure the non-Gaussianity of aprocess, for example, the fourth-order cumulant of a Gaussian process isidentically zero. On the other hand, a binary sequence of equalprobability ±1's has a fourth-order cumulant of −1 and variance ofunity.

When multiple independent processes are added together, the resultingprocess tends to approach a Gaussian process because the individualprobability density functions (PDFs) of the individual processes areconvolved together to give the PDF of the resulting process. Therefore,it may be possible to adjust a linear beamformer weight vector so thatthe beamformer output will be as non-Gaussian as possible. The utilityfunction to adjust the linear beamformer should:

Measure the deviation from non-Gaussianity

It should be scale invariant (i.e., amplification of the beamformeroutput should not change the non-Gaussianity measure).

One such utility function is the fourth-order cumulant strength:${{f(w)} = \frac{{{cum}\left( {{g^{*}(t)},{g(t)},{g^{*}(t)},{g(t)}} \right)}}{E^{2}\left\{ {{g(t)}}^{2} \right\}}},\quad {{{where}\quad {g(t)}} = {w^{H}{r(t)}}}$

It can be shown that this function is at a maximum when a source can beseparated by the linear beamformer weight vector w. A basic blockdiagram of a processor that attempts to separate a source among multiplesources based on cumulant strength is depicted in FIG. 43. Thisinterpretation provides us a way to develop a solution to the blindsignal separation problem which will be detailed in the followingsections.

21.6 Derivations for the Cumulant Recovery (CURE) System:

In this section, we provide two different approaches to derive thecumulant recovery (CURE) system which attempts to extract one of themultiple cochannel sources illuminating an antenna array.

The first approach stems from the cumulant strength maximizationconcept, we may try to evaluate the derivative of the cumulant strengthutility function with respect to the linear beamformer weight vector wand equate the result to zero and solve for w. The second approachinvolves the evaluation of cumulant matrices and their principaleigenvalues and vectors by a method that will be defined as theconstrained double-power method. The first method does not provideconvergence properties, but its equivalence with the second approachwill help us examine the convergence properties in the next section.

21.6.1 CURE Algorithm as a Cumulant Strength Maximizer:

In the following our goal is to maximize the cumulant strength utilityfunction:${{f(w)} = \frac{{{cum}\left( {{g^{*}(t)},{g(t)},{g^{*}(t)},{g(t)}} \right)}}{E^{2}\left\{ {{g(t)}}^{2} \right\}}},\quad {{{where}\quad {g(t)}} = {w^{H}{r(t)}}}$

Let us take the derivative of the above utility function with respect tothe weight vector and set the result equal to zero. The cumulant can beexpressed as:

cum(g*(t),g(t),g*(t),g(t))=E{[w ^(H) r(t)r ^(H)(t)w] ²}−2E{w ^(H) r(t)r^(H)(t)w}−E{[w ^(H) r(t)]² }E{[r ^(H)(t)w] ²}

Then, its derivative can be calculated by treating w^(H) as the variableand w as a constant: $\begin{matrix}{{\frac{\partial}{\partial w^{H}}\quad {{cum}\left( {{g^{*}(t)},{g(t)},{g^{*}(t)},{g(t)}} \right)}} = \quad {{2E\left\{ {w^{H}{r(t)}{r^{H}(t)}{{wr}(t)}{r^{H}(t)}w} \right\}} -}} \\{\quad {{4E\left\{ {{r(t)}{r^{H}(t)}w} \right\}} -}} \\{\quad {2E\left\{ {\left( {w^{H}{r(t)}} \right){r(t)}} \right\} E\left\{ \left\lbrack {{r^{H}(t)}w} \right\rbrack^{2} \right\}}}\end{matrix}$

This expression can be put in a more convenient form as follows:$\begin{matrix}{d = {\frac{\partial}{2{\partial w^{H}}}\quad {{cum}\left( {{g^{*}(t)},{g(t)},{g^{*}(t)},{g(t)}} \right)}}} \\{= {{E\left\{ {{{g(t)}}^{2}{g^{*}(t)}{r(t)}} \right\}} - {2E\left\{ {{r(t)}{g^{*}(t)}} \right\}} - {E^{*}\left\{ {g^{2}(t)} \right\} E\left\{ {{r(t)}{g(t)}} \right\}}}} \\{d_{k} = {\frac{\partial}{2{\partial w^{H}}}\quad {{cum}\left( {{g^{*}(t)},{g(t)},{g^{*}(t)},{g(t)}} \right)}}} \\{{= {{cum}\left( {{g^{*}(t)},{g(t)},{g^{*}(t)},{r_{k}(t)}} \right)}},{1 \leq k \leq {M.}}}\end{matrix}$

Using the above result for the derivative of the cumulant expression,the derivative of the utility function can be calculated as:${\frac{\partial}{\partial w^{H}}\quad {f(w)}} = {\frac{{{{sign}\left( \gamma_{4,g} \right)} \cdot \left( \sigma_{g}^{2} \right)^{2} \cdot 2 \cdot d} - {{2 \cdot {\gamma_{4,g}} \cdot E}\left\{ {{r(t)}{r^{H}(t)}w} \right\}}}{\left( \sigma_{g}^{2} \right)^{4}} = 0}$

Equating the derivative to zero, we obtain the following necessarycondition for the weight vector w to be at an extreme of the cumulantstrength utility function:${{\gamma_{4,g} \cdot E}\left\{ {{r(t)}{r^{H}(t)}} \right\} w} = {\left. {\left( \sigma_{g}^{2} \right)^{2} \cdot d}\rightarrow w \right. = {\frac{\gamma_{4,g}}{\left( \sigma_{g}^{2} \right)^{2}}\quad R^{- 1}d}}$

This equation is called as the covariance inversion CURE iteration. Ingeneral, inverting the covariance matrix results in numerical problemsif the signal to noise ratios are high. Therefore, a practical way ofdoing the above computation is to evaluate the eigendecomposition of thecovariance matrix R, and use it in place of the covariance matrixinverse as follows:$w = {\frac{\gamma_{4,g}}{\left( \sigma_{g}^{2} \right)^{2}}\quad E_{ss}\Lambda_{ss}^{- 1}E_{ss}^{H}d}$

This equation is called as the eigenCURE iteration in the measurementspace. Since the scale factors are not important for the output signalto noise ratio performance, we can compute the weight, without using thescalar $\frac{\gamma_{4,g}}{\left( \sigma_{g}^{2} \right)^{2}}.$

Then, the whole process can be summarized as:

Using a weight vector w, obtain a signal g(t):

g(t)=w^(H)r(t)

Using the signal g(t), calculate the cumulant vector d:

d_(k)=cum(g*(t), g(t),g* (t), r_(k)(t)), 1≦k≦M.

Calculate the new weights as:

w=E_(SS)Λ_(SS) ⁻¹E_(SS) ^(H)d

Obtain a new signal g(t) using the new weights (after a possiblenormalization), and continue to iterate.

It is possible to compute the necessary weight vectors if the originalmeasurements r(t) are preprocessed by a transformation matrix so thatthe dimensionality is reduced from M to P and the preprocessing is donein such a way that the resulting measurements have an identitycovariance matrix:

y(t)=T^(H)r(t), T=E_(SS)Λ_(SS) ^(−½)

E{y(t)y^(H)(t)}=I_(p)

Then, the zero derivative condition translates itself to the following:

g(t)=v^(H)y(t)

d_(k)=cum(g*(t),g(t),g*(t),y_(k)(t)), 1≦k≦P.

v=d

in which v is the weight vector in the reduced dimensionality space.Therefore, the signal separation process using the preprocessedmeasurements can be summarized as:

Using the eigenstructure of the covariance matrix, preprocess themeasurements:

y(t)=T^(H)r(t), T=E_(SS)Λ_(SS) ^(−½)

Using a weight vector v, obtain a signal g(t):

g(t)=w^(H)y(t)

Calculate the new weights as:

v_(k)=cum(g*(t),g(t),g*(t),y_(k)(t)), 1≦k≦P.

Obtain a new signal g(t) using the new weight vector v (after a possiblenormalization), and continue to iterate.

Summarizing, we have shown how the derivative of the cumulant strengthutility function can be computed both in the measurement space and thesignal subspace. We proposed iterative methods that force thisderivative to zero, and thereby find extremal points of the utilityfunction. The issue of algorithm convergence is covered below. Toaddress convergence, we examine the structure of the cumulants of themeasurements in detail below.

21.6.2 CURE Algorithm as Structured Eigenvector Computer:

In this section, we first introduce a cumulant matrix C of size P²×P² ofthe preprocessed array measurements. Using C, we describe thelimitations of the previous cumulant based signal separation methods.Finally, using the structure in C we propose a cumulant based signalseparation method that results in the maximization of cumulant strength.

21.6.2.1 Cumulant Matrix:

In this section, we introduce a P²×P² (here we assume the number ofsources and its estimate are identical) cumulant matrix C:

C(P·(i−1)+j,P·(k−1)+l)=cum(y _(i)*(t),y _(j)(t),y _(k)(t), y_(l)*(t))1≦i,j,k,l≦P

in which the measurements that are used to compute the C matrix are theoutput of the preprocessor:

y(t)=T^(H)r(t), where TE_(S)(Λ_(S)−σ_(n) ²I)^(−½)=US⁻¹

With finite samples, we can estimate the cumulant matrix C as follows:${C\left( {{{P \cdot \left( {i - 1} \right)} + j},{{P \cdot \left( {k - 1} \right)} + l}} \right)} = {{\frac{1}{N}\quad {\sum\limits_{t = 1}^{N}\quad {{y_{i}^{*}(t)}{y_{j}(t)}{y_{k}(t)}{y_{l}^{*}(t)}}}} - {\frac{1}{N^{2}}\quad {\sum\limits_{t_{1} = 1}^{N}\quad {{y_{i}^{*}\left( t_{1} \right)}{y_{j}\left( t_{1} \right)}{\sum\limits_{t_{2} = 1}^{N}\quad {{y_{k}\left( t_{2} \right)}{y_{l}^{*}\left( t_{2} \right)}}}}}} - {\frac{1}{N^{2}}\quad {\sum\limits_{t_{1} = 1}^{N}\quad {{y_{i}^{*}\left( t_{1} \right)}{y_{k}\left( t_{1} \right)}{\sum\limits_{t_{2} = 1}^{N}\quad {{y_{j}\left( t_{2} \right)}y_{l}^{*}\left( t_{2} \right)}}}}} - {\frac{1}{N^{2}}\quad {\sum\limits_{t_{1} = 1}^{N}\quad {{y_{i}^{*}\left( t_{1} \right)}{y_{l}^{*}\left( t_{1} \right)}{\sum\limits_{t_{2} = 1}^{N}\quad {{y_{j}\left( t_{2} \right)}{y_{k}\left( t_{2} \right)}}}}}}}$1 ≤ i, j, k, l ≤ P

Let the pair (γ_(4,m), σ_(m) ²) denote the fourth order cumulant andpower of the mth source respectively. Using the statistical independenceof the signals illuminating the array and the properties of fourth ordercumulants, it can be proved that the cumulant matrix C can be decomposedas:${C = {\sum\limits_{m = 1}^{P}{{\gamma_{4,m}\left( {b_{m}^{*} \otimes b_{m}} \right)}\left( {b_{m}^{*} \otimes b_{m}} \right)^{H}}}},{{in}\quad {which}},\quad {b_{m} = {T^{H}a_{m}}}$

in which ‘ ⊗ ’

denotes the Kronecker product. The P dimensional vectorb_(m)=T^(H)a_(m), denotes the effective steering vector of the mthsource after the transformation:${y(t)} = {{{\sum\limits_{k = 1}^{P}\quad {T^{H}a_{m}{s_{m}(t)}}} + {T^{H}{n(t)}}} = {{\sum\limits_{k = 1}^{P}{b_{m}{s_{m}(t)}}} + {T^{H}{n(t)}}}}$

If we recall the singular value decomposition AR_(SS) ^(½)=USV, then wehave the following relation:${b_{m} = {\frac{1}{\sigma_{m}^{2}}\quad v_{m}}},{1 \leq m \leq {P.}}$

and therefore, we can rewrite C as: $\begin{matrix}{C = {\sum\limits_{m = 1}^{P}{{\gamma_{4,m}\left( {b_{m}^{*} \otimes b_{m}} \right)}\left( {b_{m}^{*} \otimes b_{m}} \right)^{H}}}} \\{= {\sum\limits_{m = 1}^{P}{\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad \left( {v_{m}^{*} \otimes v_{m}} \right)\left( {v_{m}^{*} \otimes v_{m}} \right)^{H}}}}\end{matrix}$

Since the v_(m)'s are orthogonal to each other, the eigendecompositionof the Hermitian matrix C can be found rather easily. Its eigenvectorsare:e_(m) = (σ_(m)²)²(b_(m)^(*) ⊗ b_(m)) = (v_(m)^(*) ⊗ v_(m)), 1 ≤ m ≤ P.

with the corresponding eigenvalues:$\lambda_{m} = \frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}$

We observe that modulus of the mth principal eigenvalue is a measure ofthe strength of non-Gaussianity of the mth source, and it is a scaleinvariant quantity (multiplying S_(m) (t) with any nonzero constant willnot change |λ_(m)|).

From this analysis, we determine that if the eigenvalues of C are alldistinct (in other words, if they have different strength ofnon-Gaussianity), then the eigendecomposition of C is unique, and theprincipigenvectors reveal the modified steering vectors b_(m) in aKronecker product form(e_(m) = (σ_(m)²)²(b_(m)^(*) ⊗ b_(m)), 1 ≤ m ≤ P.)  .

However, if there are sources of the same modulation type their cumulantstrengths will be identical and the cumulant matrix C will haverepetitive eigenvalues.

Let us define an operator “unvec(·)” that converts a P²− vector h to aP×P square matrix G:

G=unvec(h)→G(i,j)=h(M·(j−1)+i)1≦m≦P

Therefore, after performing an eigendecomposition on C, we obtain e_(m)and use this vector to estimate the mth modified steering vector:${G_{m}\left( {i,j} \right)}\overset{\Delta}{=}{{{unvec}\left( e_{m} \right)} = {{{{unvec}\left( {\sigma_{m}^{2}\left( {b_{m}^{*} \otimes b_{m}} \right)} \right)}\quad 1} \leq m \leq P}}$

which yields

G_(m)=σ_(m) ²b_(m)b_(m) ^(H)=v_(m)v_(m) ^(H).

This result indicates that the columns of the unitary matrix V (i.e.,v_(m)'s) can be obtained as the principal component of G_(m) within aphase ambiguity (if v_(m) is the principal component of G_(m), then sois v_(m) exp(j2πθ) with 0 θ<1). It also indicates the limitations ofblind steering vector estimation: since both source powers and modifiedsteering vectors b_(m)'s are unknown, we cannot not identify both ofthem uniquely without a scale factor ambiguity. Furthermore, theeigendecomposition of C will yield an ordering of the eigenvectors(e_(m)'s) with respect to the magnitude of their eigenvalues(γ_(4,m)/(σ_(m) ²)²). Therefore, the unitary matrix V is determined upto an ordering of its columns in addition to a phase ambiguity, whichcan be expressed as

{tilde over (V)}=V D_(S) ^(H)P_(S) ^(H)

where P_(S) is a P×P permutation matrix that describes the reorderingencountered during the eigendecomposition of C. The P×P diagonal matrixD_(S) contains phase ambiguity factors associated with the columns ofthe unitary matrix {tilde over (V)}.

Using {tilde over (V)}, we can estimate the source waveforms as

{tilde over (V)}^(H)r(t)=P_(S)D_(S)ŝ ;_(c)(t)

which implies a reordering and phase rotation for the normalized sourcewaveform estimates and it is equivalent to obtaining ŝ ;_(c)(t) forsignal separation purposes. In addition, we can find the steeringvectors as

E_(S)(Λ_(S)−σ_(n) ² I)^(½) {tilde over (V)}=A D _(S) ^(H) P _(S) ^(H).

The steering vectors A are needed for retrieving directionalinformation. The presence of phase ambiguities and reordering does notintroduce a problem in the single block analysis, but for theblock-by-block analysis of array measurements, continuity constraintsbetween blocks will be necessary in order to prevent random (spurious)modulation of separated signals due to D_(S.) and change of theirordering (port switching) due to P_(S). This issue will be discussed inlater sections.

21.6.2.2 Limitations of Previous Cumulant-based Approaches:

We note that during this cumulant-based signal separation procedure, twolevels of eigendecompositions are required:

1. The eigendecomposition of the P² by P² cumulant matrix C to determinethe P by P, G_(m) matrices (there are P of them);

2. Principal component analysis of G_(m) matrices for each of the Psources that contribute to the measurements.

When we also consider the eigendecomposition performed on the covariancematrix before cumulant computations, the demand becomes intense forreal-time applications.

An even more important problem arises when there are sources of the sametype, which result in identical eigenvalues (γ_(4,m)/(σ_(m) ²)²). Let usconsider the presence of two identical sources (P=2). Then the 4 by 4cumulant matrix C will be rank two, but since the eigendecomposition isnot unique in this case (any linear combination of the two principaleigenvectors will be a valid eigenvector), the matrices G₁ and G₂ willboth be rank two, and their principal components will not yield thesteering vectors of interest. In the presence of more identical sources,the situation gets more complicated. This is a major problem forpreviously existing cumulant based blind signal separation algorithms(see for example the following two papers: 1) J. F. Cardoso and A.Souloumiac, “Blind Beamforming for non-Gaussian Signals,” IEEProceedings Part F, vol. 140, no. 6, pp. 362-370, December 1993, and 2)L. Tong, J. Inouye and R. Liu, “Waveform Preserving Blind Identificationof Multiple Independent Sources,” IEEE Trans. on Signal Processing, vol.41, no. 7, pp. 2461-2470, July 1993).

We end this section indicating the high level of redundancy incorporatedin the structure of the cumulant matrix C. For practical applications,it is important to use the symmetries in the cumulant matrix to reducethe computations. In the next section, we describe an approach toovercome the problems related to the eigendecomposition of C, and thecomplications originating from the presence of similar modulations. Inthe pipeline structure, computation of C is the major load in theprocessing, and to ensure a forward looking processing structure,multiple parallel processors may be dedicated to compute C, although weshow a single unit as the cumulant matrix computer.

21.6.2.3 Cumulant-based Iterative Blind Separation:

We introduce the multiple port signal recovery unit as a solution tofinding the eigenvectors of the cumulant matrix C using the powermethod. Since we know the structure of the eigenvectors (the Kroneckerproduct structure): e_(m) = (σ_(m)²)²(b_(m)^(*) ⊗ b_(m)),  1 ≤ m ≤ P.

We should exploit this information in order to

Reduce the computations

Eliminate the problems associated with the presence of sources withidentical statistics.

At each iteration, the power method computes the following, using the P²by P² cumulant matrix C: f = C(w^(*) ⊗ w)

and using w as a substitute for one of the columns of the unitary matrixV.

After proper normalization, the result f will be used to repeat thismatrix-vector multiplication until some form of convergence criterion ismet (w will be parallel to one of the v_(m)'s at convergence). Insteadof an arbitrary P²-vector, we use (w^(*) ⊗ w),

since we know that the eigenvectors must be of this form. Thismultiplication will result in $\begin{matrix}{f = {C\left( {w^{*} \otimes w} \right)}} \\{= {\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad \left( {v_{m}^{*} \otimes v_{m}} \right)\left( {v_{m}^{*} \otimes v_{m}} \right)^{H}\left( {w^{*} \otimes w} \right)}}} \\{= {\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad \left( {v_{m}^{*} \otimes v_{m}} \right){{{v_{m}^{H}w}}^{2}.}}}}\end{matrix}$

For the next iteration of the power method, we need to obtain a P-vectorfrom f that is a P² vector. Using the unvec(·) operation yields$F = {{{unvec}(f)} = {\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad \left( {{v_{m}^{H}v}}^{2} \right)v_{m}v_{m}^{H}}}}$

We observe that the eigenvectors of F are the columns of the unitarymatrix V (v_(m)'s). Since our substitute for v_(m) is w, we can reusethis weight vector for the power method on F:${Fw} = {\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad {{v_{m}^{H}v}}^{2}\left( {v_{m}^{H}w} \right)v_{m}}}$

Therefore, the overall iteration can be thought of a cascade of twopower method iterations, and it can be described asw_(b + 1) = α_(b + 1) ⋅ unvec(C(w_(b)^(*) ⊗ w_(b)))w_(b)

where the constant a_(b+1) is chosen so that the norm of w_(b+1) isunity.

Using cumulant properties, it is possible to perform the iterationwithout computing the cumulant matrix C and performing unvec(·)operation. To see this, we rewrite the iteration as${w_{b + 1}(j)} = {\alpha_{b + 1} \cdot {\sum\limits_{i,k,{l = 1}}^{P}\quad {{cum}\left( {{y_{i}^{*}(t)},{y_{j}(t)},{y_{k}(t)},{y_{l}^{*}(t)}} \right)}}}$w_(b)(i)w_(b)^(*)(k)w_(b)(l), 1 ≤ j ≤ P.

Using the multilinearity property of the cumulants, followed by thepermutation property, we obtain${w_{b + 1}(j)} = {\alpha_{b + 1} \cdot {\sum\limits_{i,k,{l = 1}}^{P}\quad {{cum}\left( {{{w_{b}(i)}{y_{i}^{*}(t)}},{{w_{b}^{*}(k)}{y_{k}(t)}},{{w_{b}(l)}{y_{l}^{*}(t)}},{y_{j}(t)}} \right)}}}$

Finally, using the additivity property of cumulants,${w_{b + 1}(k)} = {\alpha_{b + 1} \cdot \quad {{{cum}\left( {{\sum\limits_{i = 1}^{P}{{w_{b}(i)}{y_{i}^{*}(t)}}},{\sum\limits_{k = 1}^{P}{{w_{b}^{*}(k)}{y_{k}(t)}}},{\sum\limits_{l = 1}^{P}{{w_{b}(l)}{y_{l}^{*}(t)}}},{y_{j}(t)}} \right)}.}}$

Noting that $\sum\limits_{k = 1}^{P}{{w_{b}^{*}(k)}{y_{k}(t)}}$

is the inner product w_(b) ^(H)y(t), we have

w _(b+1)(j)=α_(b+1)·cum (g _(b)*(t),g _(b)*(t), g _(b)*(t),y_(j)(t)),1≦j≦P

where the waveform gb(t) is obtained as

g_(b)(t)=w _(b) ^(H)(t)y(t).

With true cumulants, the weight vector w will satisfy the followingrelationship at convergence:$w = {{\alpha \cdot {\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad {{v_{m}^{H}w}}^{2}\left( {v_{m}^{H}w} \right)v_{m}}}} = {\alpha \quad {R_{c}(w)}{w.}}}$

The matrix R_(c)(w)has the same structure as an array covariance matrixR, if we consider (γ_(4,m)/(σ_(m) ²)²)|w^(H)v_(m)|² as source powers.The introduction of weight vector w enables us to tune the effectivesource powers in R_(c)(w), using the cumulants of processedmeasurements. For this reason, we named the proposed signal separationmethod the Cumulant-Recovery (CURE) algorithm. The idea of using thepreprocessed signals provided by a transformation obtained from theeigendecomposition of the covariance matrix finalizes the name as theeigenCURE algorithm.

Solutions to the convergence relationship are of the main interest here,and we note that they include the set of vectors v₁, v₂, . . . , v_(p)which are the target vectors for source separation. To see this, we notethat v_(k) ^(H)v_(m)=δ_(k,m), and let w=v_(k), which yields${{R_{c}\left( v_{k} \right)}v_{k}} = {{\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad \delta_{k,m}\quad v_{m}}} = {{\frac{\gamma_{4,k}}{\left( \sigma_{k}^{2} \right)^{2}}\quad v_{k}} = {\alpha \quad {v_{k}.}}}}$

21.6.2.4 Direct versus Beamforming-based Computation of the CumulantIteration:

In this section, we summarize beamforming and direct computationapproaches for cumulant iteration.

21.6.2.5 Cumulant Iteration Using Beamforming:

In this section, we briefly describe the fundamentals of a single portoperation for the capture of a single source among P sources.

1. Transformation of Steering Vectors: Project the steering vectorestimate a_(init), onto the reduced dimensional space by thetransformation matrix T, provided by normalization:${b_{init} = {T^{H}a_{init}}},{w = \frac{b_{init}}{b_{init}}}$

2. Beamforming: Using the projected and normalized steering vector w,combine the preprocessed measurements y(t) and form the auxiliary signal

g(t): g(t)=w^(H)y(t)

3. Computation of CrossCumulant Vector: Using the auxiliary signal andprojected measurements y(t), compute the cumulant vector:

b_(j) 32 cum(g*(t),g(t),g*(t),y_(j)(t)), 1≦j≦P

and normalize its norm: $b = \frac{b}{b}$

4. Orthogonalization of Cross-Cumulant Vector: Using Gram-Schmidt andPort Priorities, orthogonalize the cross-cumulant vector with respect tocross-cumulant vectors of sources with higher priority. Obtain a unitnorm vector d by processing b.

5. Continue Iterations: Using the unit norm d vector obtained in step 4,as a new weight vector, go to step 2, and use beamforming to obtain anew auxiliary signal. Repeat Steps 2 through 5 until convergencecriteria are met.

6. Backprojection: Use the final estimates of the d vector from Step 4after convergence, use backprojection to estimate the steering vectorsin the measurements space using the eigenstructure of the covariancematrix for the current block:

â ;=E _(S)(Λ_(S)−{circumflex over (θ)}² I _(p))^(+½)d

7. Signal Extraction and Steering Vector Propagation: Use the steeringvector estimate â ; for recovering the source waveform for the currentblock and as an initial estimate of the next block's adaptation, i.e.,for the current block, set:

a_(init)=â ;.

21.6.2.5 Direct Cumulant Iteration:

In this section, we provide a way to compute the cross cumulant vectorb, defined as:

b_(j)=cum(g*(t),g(t),g*(T),y_(j)(t)), 1≦j≦P

in Step 3 of the previous section, using a P² by P² cumulant matrix C.

The matrix under consideration is defined as:

C(P·(i−1)+j,P·(k−1)+l)=cum(y _(i)*(t),y _(j)(t),y _(k)(t),y _(l)*(t))1≦i,j,k,l≦P

Now, let us consider the following matrix vector multiplication thatinvolves Kronecker products:

b=(w*I)^(H)C(w*w)

Using the definition of the C matrix, and the definition of matrixmultiplication and Kronecker product:${b_{j} = {\sum\limits_{i,k,{l = 1}}^{P}\quad {{{cum}\left( {{y_{i}^{*}(t)},{y_{k}(t)},{y_{i}^{*}(t)},{y_{j}(t)}} \right)} \cdot \left\lbrack {w_{i}w_{k}^{*}w_{l}} \right\rbrack}}},\quad {1 \leq j \leq P}$

Using the multilinearity property of the cumulants:$b_{j} = {\sum\limits_{i,k,{l = 1}}^{P}\quad {{cum}\left( {{w_{i}{y_{i}^{*}(t)}},{w_{k}^{*}{y_{k}(t)}},{w_{l}{y_{i}^{*}(t)}},{1 \cdot {y_{j}(t)}}} \right)}}$

Finally, using the additivity property of cumulants,$b_{j} = {{cum}\left( {{\sum\limits_{i = 1}^{P}{w_{i}{y_{i}^{*}(t)}}},{\sum\limits_{k = 1}^{P}{w_{k}^{*}{y_{k}(t)}}},{\sum\limits_{l = 1}^{P}{w_{l}{y_{i}^{*}(t)}}},{y_{j}(t)}} \right)}$

Noting that $\sum\limits_{k = 1}^{P}{w_{k}^{*}{y_{k}(t)}}$

is the inner product w^(H)y(t), we have

b_(j)=cum(g*(t),g(t),g*(t),y_(j)(t)), 1≦j≦P

where the waveform g(t) is obtained as

g(t)=w^(H)(t)y(t).

The last two equations prove that:b = (w^(*) ⊗ I)C(w^(*) ⊗ w) ⇔ b_(j) = cum(g^(*)(t), g(t), g^(*)(t), y_(j)(t)),   1 ≤ j ≤ P.

This is the equivalence of direct computation to the beamformingapproach. It is evident that the b vector is identical to the d vectorfor the cumulant strength maximization approach; both methods areidentical except for the transformation matrix T.

21.6.3 Reference Signal Exploitation Framework for the CURE Blind SignalSeparator:

In this section, we provide a framework that enables us to visualize theCURE algorithm as a reference signal based processor that uses thecross-cumulants of the reference signal that is generated from the arraymeasurements in order to estimate the steering vectors of sources andtheir waveforms.

Using this framework, we investigate the relationship of the CUREalgorithm with the CMA and SCORE algorithms that exploit properties ofsignals of interest. We determine that CMA and SCORE are special casesof the CURE algorithm if desired signal properties are incorporated inthe reference signal that CURE uses to compute cross-cumulants. Based onthis observation, we propose extensions of the CURE algorithm to useproperties of the desired signals more efficiently to achieve signalseparation.

21.6.3.1 Reference Signal Exploitation Framework:

Conventional adaptive beamformers use a reference signal to determinethe steering vector of a desired signal, i.e.,

â ;=E{r(t)g*(t)}

in which â ; is the estimate of the steering vector, r(t) is the Mchannel measurement vector, and g(t) is the reference signal which isonly correlated with the signal of interest and uncorrelated with theother sources that contribute to the measurements. After the steeringvector estimate is available, we can determine the weight vector usingthe MVDR beamformer:

w=βR⁻¹â ;

where β is a constant that does not change the signal to interferenceratio, but can control the power of the recovered signal. The waveformestimate can be obtained as follows:

ŝ ;_(d)(t)=w^(H)r(t)

If the original measurement vector is projected onto the signal subspaceusing the transformation matrix T=E_(S)Λ_(S) ^(−½):

y(t)=T^(H)r(t), E{y(t)y^(H)(t)}=I_(p)

where P is the number of sources. Then, using a reference signal g(t),the steering vector estimate for the reduced dimensional space, b(P-vector), can be obtained as:

{circumflex over (b)}=E{y(t)g*(t)}

and the desired source waveform can be obtained using ŝ;_(d)(t)=βb^(H)r(t), since the covariance matrix of reduced dimensionalmeasurements is the identity matrix, i.e.,

E{y(t)y^(H)(t)}=I_(p).

Now, let us consider the CURE iteration, in which we provide theestimate of the steering vector b for the reduced dimensionalmeasurements using a cross-cumulant vector computation:

{circumflex over (b)}=E{g*(t)g*(t)g*(t)y(t)}−E{| g(t)|²}E{y(t)g*(t)}−E*{g ²(t)}E{y(t)g*(t)}, with g(t)=v ^(H) y(t)

in which the weight vector v can be chosen to have unit norm, i.e.,∥v∥=1, so that the signal g(t) will have unit power, i.e., E{|g(t)|²}=1,which results in simplifications in the computation of {circumflex over(b)}.

Now, let us step back from the cumulant-based steering vectorestimation, and consider the case for which we have three differentreference signals, instead of only one, i.e., we have g₁,(t),g₂(t),g₃(t)which are only correlated with the signal of interest and uncorrelatedwith the remaining part of the measurements.

If we use all of the three reference signals in the cross-cumulant-basedsteering vector estimation for the reduced dimensional space, then weneed to compute the following cross-cumulant vector:

Therefore, our cumulant-based signal separation technique, that uses aweight vector to generate a reference signal g(t) and use thecross-cumulants of g(t) and preprocessed measurements y(t) can beconsidered as a special application of the three reference signals casein which we choose to select:

g₁(t)=g₂(t)=g₃(t)=g(t)=v^(H)y(t).

From this it follows that if we have multiple guesses for the weightvector, then we can form reference signals with them and use (at mostthree of such reference signals) to calculate the steering vectorestimate {circumflex over (b)}.

Finally, once {circumflex over (b)} is computed, the steering vectorestimate â ; in the original measurement space can be computed using:

â ;=E_(S)Λ_(S) ^(½){circumflex over (b)}

21.6.3.2 Relationships with Other Algorithms:

Constant modulus algorithm (CMA) and cyclostationary-restoral (SCORE)algorithms are developed with assumptions about the signal of interest.In the development of CURE algorithm, we assumed that signals ofinterest are non-Gaussian. In CMA, signals of interest are assumed to beconstant-modulus signals, and in SCORE algorithm, it is assumed thatsignals of interest possess cycle frequencies which are known a priorior can be estimated reliably. In this section, we show how we can reducethe CURE algorithm to CMA and SCORE algorithms using assumptions on thesignal of interest.

21.6.3.2.1 CURE and CMA Relationship:

In this section, we assume that the signals of interest have constantmodulus. Therefore, we can further improve the quality of the referencesignal g(t)=v^(H)y(t), by normalizing its amplitude to be unity and thenew reference signal can be obtained as:

g_(n)(t)=g(t)/|g(t)|

We can then set g₁(t)=g₂(t)=g₃(t)=g_(n)(t), and compute thecross-cumulant vector to estimate {circumflex over (b)}:

{circumflex over(b)}=E{y(t)g*(t)/|g(t)|}−2·1·E{y(t)g*(t)/|g(t)|}−E*{g(t)g(t)|²}E{y(t)/|g(t)|}

For circularly symmetric signals, the second term in the aboveexpression is theoretically zero, and therefore the expression can besimplified as:

{circumflex over (b)}=E{y(t)g*(t)/|g(t)|}

In this last equation, we ignored the sign multiplier, because it doesnot change the signal to noise ratio of a beamformer that uses{circumflex over (b)} for beamforming. If we choose to implement thisalgorithm without projecting the measurements onto the signal subspace,then the estimate of the steering vector will be:

â ;=E{r(t)g*(t)/|g(t)|}

which is used to compute the weight vector using the inverse of thecovariance matrix R:

w=βR⁻¹â ;.

By comparing the last two equations to the LS-CMA [described in: B.Agee, “The least-Squares CMA: A New Technique for Rapid Correction ofConstant Modulus Signals,” Proc. ICASSP- 86, pp. 953-956, Tokyo, Japan,April 1986], we find that they are identical. Therefore, LS-CMA can beviewed as a version of the CURE algorithm when the desired signals areassumed to be constant modulus and circularly symmetric.

We note that LS-CMA is a wrong implementation of the cumulant algorithmfor signals that are not circularly symmetric. The reason for thisproblem is that we cannot ignore the termE*{g(t)g(t)/|g(t)|²}E{y(t)g(t)/|g(t)|} in the fourth-order cumulantexpression in such a scenario. Therefore, the results of the LS-CMA forsuch sources are not expected to be satisfactory and one example is thetwo BPSK signals with zero carrier offsets. In this case, LS-CMAcombines the sources to form a 4-QAM type of signal which is constantmodulus and circular y symmetric.

Finally, we note that intermediate algorithms can be developed bychoosing different reference signals for the cross-cumulant computation.For example, e may choose g₁(t)=g₂(t)=g(t)=v^(H)y(t), andg₃(t)=g_(n)(t), and obtain a steering vector estimate using:

{circumflex over (b)}=E{g*(t)|g(t)|y(t)}−E{|g(t)|²}E{y(t)g*(t)/|g(t)|}−E{|g(t)|}E{y(t)g*(t)}−E*{g(t)g(t)/|g(t)|}E{y(t)g(t)}

If the signal recovered by the weight vector v is circularly symmetricand v is unit norm, we can simplify the expression:

{circumflex over(b)}=E{g*(t)|g(t)|y(t)}−E{y(t)g*(t)/|g(t)|}−E{|g(t)|}E{y(y(t)g*(t)}

1.6.3.2.2 CURE and SCORE Relationship:

In CURE algorithm, we obtain a reference signal g(t) by using the linearbeamformer and use this reference signal to update the weights of thebeamformer through a cumulant vector computation. In certain scenarios,frequency-shifted and delayed versions of the desired signal iscorrelated with itself, and if we want CURE algorithm to extract thewaveform from such a source, we can make use of the modified referencesignal g_(n)(t):

g _(n)(t)=g(t−τ)exp (j2παt)=v ^(H) y(t−τ)exp(j2παt)

We can compute the cumulant vector using g_(n)(t):

{circumflex over (b)}_(l)=cum(g_(n)*(t),g_(n)(t),g_(n)*(t),y_(l)(t)),1≦l≦M.

Now, in an effort to reduce the computations, we can use the fact thatthe phase of g_(n)(t) is very similar to that of the desired signal nearconvergence and that the information content of g_(n)(t) may very wellbe represented by its phase (amplitude is assumed to carry relativelysmall information): we can use h(t)=g_(n)(t)/|g_(n)(t)|, as thereference signal:

{circumflex over (b)}_(l)=cum(h*(t),h(t),h*(t),y_(l)(t), 1≦l≦M.

For circularly symmetric signals, this approach results in a hybrid formwhich has relations to CMA and SCORE (scale factors are ignored):

{circumflex over (b)}=E{y(t)g*(t−τ) exp(−j2παt)/|g(t−τ)|}

which possesses the capture effect of CMA in the case of multiplecyclostationarity sources with the same cycle frequency and signalselectivity property of SCORE for resistance against noise capture.

Finally, we note that a specific mix of reference signals g_(n)(t) andh(t)=g_(n)(t)/|g_(n)(t)|, in the cumulant vector

{circumflex over (b)}₁=cum(g_(n)*(t), g_(n)(t),h*(t),y_(l)(t)), 1≦l≦M.

gives the power-SCORE iteration [B. G. Agee, S. V. Schell, and W. A.Gardner, “Spectral Self-Coherence Restoral: A New Approach to BlindAdaptive Signal Extraction Using Antenna,” Proceedings of the IEEE, vol.78, pp. 753-767, April 1990]:

{circumflex over (b)}=E{y(t)g*(t−τ)exp(−j2παt)}

when we ignore the scale factors and assume circular symmetry.

21.7 Convergence of the Blind Signal Separator:

In this section, we analyze the convergence characteristics of the blindsignal separator based on cumulants. We also provide optional methodsfor assessment of convergence, and initialization methods for betterinitialization of the iterative algorithm.

21.7.1 Superexponential Convergence:

We now analyze the convergence characteristics of the cumulant-basediterations. Let us assume that the initial weight vector w₀ isarbitrarily chosen (but has unit norm) and can be expressed as$w_{0} = {\sum\limits_{k = 1}^{P}\quad {{\beta_{k}(0)}v_{k}}}$

since the set v₁, . . . ,v_(p) forms an orthonormal basis for the Pdimensional space. Since w₀ has unit norm, we have${\sum\limits_{k = 1}^{P}\quad {{\beta_{k}(0)}}^{2}} = 1.$

After an iteration, we obtain w₁ as:$w_{1} = {\alpha_{1}\quad {\sum\limits_{k = 1}^{P}\quad {\underset{\underset{\beta_{k}{(1)}}{}}{\frac{\gamma_{4,k}}{\left( \sigma_{k}^{2} \right)^{2}}\quad {{\beta_{k}(0)}}^{2}\beta_{k}^{*}(0)}v_{k}}}}$

where α₁ is a scale factor to make the norm of w₁ be equal to unity.

The last expression is derived using the orthogonality of {v_(k)}_(k=l)^(P). We can express the weight vector after the Lth iteration in thefollowing form:$w_{L} = {\alpha_{L}{\sum\limits_{k = 1}^{P}\quad {{\beta_{k}(L)}v_{k}}}}$

where α_(L) is a scale factor to make the norm of w_(L) be equal tounity, i.e.,${\sum\limits_{k = 1}^{P}\quad {{\beta_{k}(L)}}^{2}} = {1/{\alpha_{L}^{2}.}}$

The coefficients {β_(k)(L)}_(k=1) ^(P) determine the decomposition ofthe weight vector the Lth iteration in terms of the basis vectors{v_(k)}_(k=1) ^(P). At convergence, we want the modulus of only one ofthe b_(k)'s to be unity and the rest be all zeros (since w has to beunit norm). We can obtain the recursive relation${\beta_{k}(L)} = {\frac{\gamma_{4,k}}{\left( \sigma_{k}^{2} \right)^{2}}\quad {{\beta_{k}\left( {L - 1} \right)}}^{2}{\beta_{k}^{*}\left( {L - 1} \right)}}$

To determine the signal contributions, we are interested in the norm ofβ_(k)(L). From the recursive relation, we obtain${{\beta_{k}(L)}} = {{\frac{\gamma_{4,k}}{\left( \sigma_{k}^{2} \right)^{2}}}^{({3^{L_{+}1} - 3})}{{\beta_{k}(0)}}^{3^{L}}}$

Note that the norm of β_(k)(L)grows superexponentially unlike otheralgorithms that grow in an exponential fashion.

Let us compare the ratio of the contributions from two sources (amultiple-source case can be treated in the same way); we define${c(L)}\overset{\Delta}{=}{{{\beta_{1}(L)}/{\beta_{2}(L)}}}$

and$\gamma \overset{\Delta}{=}{{{\left( {\gamma_{4,1}/\gamma_{4,2}} \right)\left( {\alpha_{2}^{4}/\alpha_{1}^{4}} \right)}}.}$

We have the following recursive relation:

c(L)=γ[c(L−1)]³=γ³ ^(L+1) _(−3)/6)[c(0)]³ ^(L)

As L increases c(L) may tend to +∞, zero, or a finite positive number.If c(L)→+∞, then this implies the weight vector converges to v₁. Ifc(L)→0, then the weight vector converges to v₂. But if c(L) converges toa finite positive constant, then the weight vector will be a linearcombination of v₁ and v₂, which is not a desired result. We nowinvestigate for which values of g and c(0) this undesired situation mayhappen, as L→+∞. Take the logarithm of c(L) to obtain

(3^(L))log(c(0))+(3^(L+1)−3)/6 log(γ)=finite constant

Divide both sides by 3^(L) and let L→+∞, to obtain

2 log(c(0))+log(γ)=0

which yields the nonconvergence condition (depicted in FIG. 44.).$\gamma = {\frac{1}{{c(0)}^{2}}.}$

However, this is an unstable condition since it is a strict equalitythat has almost zero probability of occurrence; errors in the estimatedcumulants lead the solution to convergence. We also note that in thecase of sources with identical cumulant strengths (i.e., g=1),convergence is still possible if c(0)≠1 (i.e., the initial weight vectorhas different gains on different sources). This surprising result is dueto our exploitation of the structure of C using the double-power method.

21.7.2 Assessment of Convergence:

Since we are using an iterative algorithm to estimate the steeringvectors and waveforms of cochannel sources, it is necessary to have arule to stop iterations. It is possible to perform “a few” iterationseach block and continue for some applications. However, in some othercases it may be desirable to monitor convergence of the iterations, andreact in the case of bad initialization or impose conditions to speedthe convergence of the iterative process.

To accomplish our goal, we return to the conditions of convergence: Withtrue cumulants, the weight vector w will satisfy the followingrelationship at convergence:$w = {{\alpha \cdot {\sum\limits_{m = 1}^{P}\quad {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}}\quad {{v_{m}^{H}w}}^{2}\left( {v_{m}^{H}w} \right)v_{m}}}} = {\alpha \quad {R_{c}(w)}w}}$

Since the vectors v_(m)'s form an orthonormal set, at convergence,R_(c)(w) should be rank 1, and its principal eigenvector should be w.Therefore, after iterations, we can form the cumulant matrix R_(c)(w),using its definition:

R _(c)(w)=E{|g(t)|² y(t)y ^(H)(t)}−E{|g(t)|² }E{y(t)y^(H)(t)}−|E{g(t)y(t)}|²

and compute its eigenvalues. The first two eigenvalues are sufficient toidentify the convergence situation: the principal eigenvalue shoulddominate the second one. If this is the case, a further improved weightvector estimate is the principal eigenvector of the matrix R_(c)(w). Ifthe first two eigenvalues are very close to each other, then theconvergence is not established and initialization of the iterations isnot satisfactory. Then we may try to compute the first two eigenvectors,and form a random combination of them to replace the weight vector w.

Computation of eigenvalues and eigenvectors for the matrix R_(c)(w) canbe done using the power method as well. This will eliminate the need toestimate the matrix, and compute only cumulant vectors instead of acumulant matrix. For example, to compute the first eigenvector, e, wecan iteratively compute:

Beamform:{tilde over (g)}(t)=e^(H)y(t), with e unit norm.

Update: e=E{g(t)g*(t){tilde over (g)}*(t)y(t)}−E{|g(t)|²}E{{tilde over(g)}*(t)y(t)}−E{g(t){tilde over (g)}*(t)}E{g* (t)y(t)}−E{g* (t){tildeover (g)}* (t)}E{g(t)y(t)},

Normalize: e=e/∥e∥,

Start over: {tilde over (g)}(t)=e^(H)y(t)

21.7.3 Initialization Using a Simple Cumulant-based EigendecompositionMethod:

In this section, we briefly describe a simple eigendecomposition-basedsignal separator that operates on a matrix of cumulants of size P by P.We may use this simple method to roughly estimate the steering vectorsand provide these as starting points (initialization) for the iterativeapproach that will follow. In addition, it is possible to performtracking by comparing the steering vectors from the previous block bythe results for the current block of the simple method. The cumulantmatrix under consideration is defined as follows:

H(i,j)=cum(r_(k)*(t),r_(l)(t),y_(i)(t),y_(j)*(t)) 1≦i,j≦P, 1≦k, l≦M.

Note that two of the arguments came from the unprocessed measurementsand they are fixed for the whole matrix, and the remaining two vary overthe preprocessed measurements. Using the properties of cumulants and thesignal model, we can express H as below: $\begin{matrix}{H = {\sum\limits_{m = 1}^{P}{{\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}} \cdot \left( {a_{m,k}^{*}a_{m,l}a_{m}^{2}} \right) \cdot v_{m}}v_{m}^{H}}}} \\{{= {\sum\limits_{m = 1}^{P}\quad {\mu_{m} \cdot v_{m} \cdot v_{m}^{H}}}},\quad {{{where}\quad \mu_{m}} = {\frac{\gamma_{4,m}}{\left( \sigma_{m}^{2} \right)^{2}} \cdot {\left( {a_{m,k}^{*}a_{m,l}\sigma_{m}^{2}} \right).}}}}\end{matrix}$

in which α_(m,l) indicates the response of the lth sensor to the mthsource. Provided that μ_(m)'s are all different, then we can easilyconclude that the eigenvectors of the matrix H are the set of vectorsv_(m)'s (multiply H from right with v_(m)). Using an inversetransformation, it is possible to estimate the steering vectors for thesources, and using the eigenvectors of H it is possible to estimatesource waveforms.

The problems with this approach arises when some of the μ_(m)'s are veryclose to each other. There is no way to select the best possiblechannels k and l. It is quite possible that some of the sources may bereceived very poorly by these channels and their steering vectors cannot be identified accurately. Nevertheless, this approach gives usinitialization points to start the iterative algorithm of the nextsection and furthermore allows us to check whether the sources thatexisted in the previous block are still present by multiplying H withtheir projection to the signal subspace and test whether these projectedvectors are eigenvectors of H.

21.8 Implementation Details and Flow Chart:

The heart of the CURE algorithm (signal subspace computations) issummarized in the following iterative process:

Using the eigenstructure of the covariance matrix, preprocess themeasurements as follows:

y(t)=T^(H)r(t), T=E_(SS)(Λ_(SS)−{circumflex over (σ)}²)^(−½)

Using a P dimensional weight vector v, obtain a signal g(t):

g(t)=w^(H)y(t)

Calculate the new weights as:

v_(k)=cum(g*(t),g(t), g*(t),y_(k)(t)), 1≦k≦P.

Obtain a new signal g(t) using the new weight vector v (after a possiblenormalization), and continue to iterate.

Although the above steps form the core of the algorithm, there areadditional steps that are necessary to reduce the algorithm to practice.One issue is how to capture different sources. Another important one isto undo arbitrary phase and gain modulations on the recovered signals.

In the eigenCURE algorithm, multiple parallel ports are used to extractdifferent sources. To ensure that each port extracts a different source,the weight vectors in the P dimensional space are constrained to beorthogonal. The orthogonalization is achieved by a Gram-Schmidtprocessor and the order of orthogonalization is based on portpriorities. Port priority for each signal extraction port is computed asa function of the cumulant strength of the waveform it produces used asteering vector from a previous block and the current blockmeasurements.

In the pipeCURE algorithm, since we have the entire cumulant matrixprovided by the cumulant matrix computer, the iterations can be done assimple matrix vector multiplications for which commercial processorshave the built-in functions to handle very efficiently. This enables usto make multiple iterations to improve our results.

Let us assume that we have P sources, and we have an estimate of thesteering matrix denoted by Ã, which can be either the initial (startup)values or the results provided from the pipeCURE for the previous block.

The following indicates the flow of operations consistent with the blockdiagram of FIG. 7.:

1. Transformation of Steering Vectors: Project the steering matrixestimate onto the reduced dimensional space by the transformation matrixT:

{tilde over (B)}=T^(H)Ã

2. Cumulant Strength Computation: Normalize the norm of each column of{tilde over (B)} and store the results in {tilde over (V)}, and thencompute the cumulant strength for each signal extracted by the weightsusing the matrix vector multiplication:${O_{m} = {{\left( {{\overset{\sim}{v}}_{m}^{*} \otimes {\overset{\sim}{v}}_{m}} \right)^{H}{C\left( {{\overset{\sim}{v}}_{m}^{*} \otimes {\overset{\sim}{v}}_{m}} \right)}}}},\quad {{{for}\quad 1} \leq m \leq {P.}}$

3. Priority Determination: Reorder the columns of {tilde over (V)} andform the matrix {tilde over (W)} so that the first column of {tilde over(W)} yields the highest cumulant strength and the last column of {tildeover (W)} yields the smallest cumulant strength.

4. Recovery of the First Signal: Starting with the first column of{tilde over (W)}, proceed with the double-power method followed byGram-Schmidt orthogonalization with respect to higher priority columnsof {tilde over (W)}, i.e., for column k:${{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)} = {\alpha_{b + 1} \cdot {{unvec}\left( {C\left( {{{\overset{\sim}{w}}_{k}^{*}(b)} \otimes {{\overset{\sim}{w}}_{k}(b)}} \right)} \right)}}},{{\overset{\sim}{w}}_{k}(b)}$

where the constant a_(b+1) is chosen so that the norm of {tilde over(w)}_(k)(b+1) is unity and b is the iteration number. This operation isfollowed by the Gram-Schmidt orthogonalization:${{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)} = {\beta_{b + 1} \cdot \left( {{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)} - {\sum\limits_{l = 1}^{k - 1}{w_{l} \cdot \left( {w_{l}^{H}{{\overset{\sim}{w}}_{k}\left( {b + 1} \right)}} \right)}}} \right)}},{{{since}\quad {w_{l}}} = 1.}$

where the constant b_(b+1) is chosen so that the norm of {tilde over(w)}_(k)(b+1) is unity . In the last expression, w_(l) denotes the finalweight vector with source of priority l..

5. Multiple Signal Recovery: Repeat Step 4 for each column for apredetermined K times. After iterations are complete for the kth column,declare the resultant vector as w_(k), and proceed with the remainingcolumns. After all sources are separated, form the matrix W thatconsists of w_(k)'s as its columns.

6. Port Association: After all the power method and Gram-Schmidtiterations are complete, compare the angle between columns of W and thecolumns of {tilde over (V)}. This can be obtained by taking the absolutevalues of the elements of he matrix W^(H){tilde over (V)}, and take thearccosine of each component. To find the port number assigned to thefirst column of W in the previous block, simply take the index of thecolumn of the matrix W^(H){tilde over (V)}, and look for the entry withthe maximum absolute value. For the second column of W, proceed the sameway except this time we do not consider previously selected ports. Usingthis rule, reorder the columns of W, such that there is no portswitching involved. Let us denote the resultant matrix as {circumflexover (V)}.

7. Steering Vector Adjustment: Due to the blindness of the problem,estimated steering vectors are subject to arbitrary gain and phaseambiguities. The gain ambiguities are corrected by the unit amplitudeconstraint on the columns of {circumflex over (V)}. However, this doesnot prevent phase modulations on the columns of this matrix. To maintainthis continuity, we compute the inner product of each column of{circumflex over (V)} with the corresponding column in {tilde over (v)}and use the resulting scalar to undo the phase modulation, i.e.,

ε_(m)=angle({tilde over (v)} _(m) ^(H) v _(m)), and v _(m) =v_(m)·exp(−jε_(m))

8. Backprojection: In order to use the current steering vector estimatesfor the next processing block, we need to backproject the steeringvector estimates for the reduced dimensional space to the measurementspace. This yields the estimate of the steering matrix and can beaccomplished as:

Â ;=E _(S)(Λ_(S)−{circumflex over (σ)}² I _(P) _(ε) )^(+½){circumflexover (V)}

Â ; will be used in the next block as Ã as an estimate of the steeringmatrix in the first step of the multiple port signal extraction unit.

9. Beamforming: It is important to note that beamforming for P sourcesrequires a matrix multiplication of two matrices: the {circumflex over(V)} matrix that is P by P, and the reduced dimensional observationmatrix y(t), which is P by N, where N is the number of snapshots.Usually N is larger than P and this matrix multiplication may take along time because of its size. Therefore, it may be appropriate to dofinal beamforming in another processor since it does not introduce anyfeedback. Final beamforming is accomplished as:

ŝ ;(t)={circumflex over (V)}^(H)y(t)

The estimated signals will be sent to the correct post processing unitsbecause of the orderings involved.

The cumulant-based post-processing unit that estimates the steeringvectors and source waveforms for the sources in a blind fashion usinghigher order statistics of the reduced dimensional observations. The useof higher order statistics appears in the multiplications that containthe cumulant matrix C. Other than the last step of beamforming, theprocessing does not involve multiplication of large time-consumingmatrices.

21.9 Simulations:

In this section, we describe simulation experiments to illustrate theoperation of the eCURE algorithm on both stationary and nonstationaryenvironments. The eCURE algorithm has four ports to recover at most foursources at any given time. We use an eight-element linear antenna arraywith uniform (half-wavelength) spacing and identical sensors, and thesteering vectors take the form:

a(θ)=[1, exp(jπ sin (θ)), . . . , exp (jπ7 sin(θ))]^(T)

in which θ is the direction of arrival. We use the MATLAB computationalenvironment to generate and process the data.

21.9.1 Experiment 1-Non-Transient Sources:

In this experiment, we simulate a case in which there are threefar-field sources. Two of the sources are single-tone frequencymodulated (FM) signals and the third source is an amplitude modulated(AM) speech waveform. The single-tone FM signals can be described inanalytic signal representation:

s _(k)(t)=A _(k) exp(j2π(f _(k) t+β _(k) sin(2πα₁))), k=1,2

For the first source, we have

A₁=10, f₁=0.04, β₁=0.4, α₁=0.1

and for the second source, we have

A₂=10, f₂=0.02, β₂=0.3, α₂=0.07.

The directions of arrival are θ₁=−5°, and θ₂=0°. The measurement noisecomponent is Gaussian, spatially and temporally white, and has equalpower (0 dB) at all the sensors. Therefore, both FM sources have a powerlevel of 20 dB. The spectra of the FM signals is illustrated in FIGS. 45and 46. It is important to note that the envelope of the FM sources isconstant (i.e., |s₁(t)=|=s₂(t)|=100.

The AM signal is created using samples from a real speech signal. Weassume that it is centered at the center frequency of analysis and thereis no carrier (double-sideband suppressed carrier AM (DSB-SC AM)).Therefore, the analytic representation of the third signal is identicalto the speech samples. We collected 15,000 samples of the speechwaveform and scaled it in such a way that it has a power of 15 dB.However, since the amplitude of the speech waveform varies over time(unlike the FM signals), its power changes from block to block. Spectrumof the speech signal is calculated by using the samples (14,001 to15,000), which are displayed in FIG. 47.

The eCURE algorithm with a block size of 1,000 snapshots is used toanalyze the generated measurements and therefore, we have 15 blocks forthe simulation. We used the version of the eCURE in which waveformcontinuity is maintained by normalizing the waveform estimate by thefirst component of the estimated steering vector for each port. Thecapture strengths used for the eCURE are only a function of the anglesbetween the projected steering vector from the previous block(v_(k)=T^(H)a_(k)(m−1)) and the current block cumulant vector for eachport (b_(k)(m)), i.e., the capture strength for the kth port for the mthblock is |v_(k) ^(H)b_(k)(m)|/∥v_(k)∥. The number of sources detectorprovided correct estimates for all the blocks.

FIG. 48 shows the bearing estimates from the ports. As it can be seenfrom the figure, Port 1 captures the second FM source, Port 2 capturesthe AM source, and Port 3 captures the first FM source. The eCUREalgorithm does not use array calibration information but the resultsillustrated in FIG. 48 are given to indicate the fast convergence speedof the eCURE algorithm to the correct steering vectors in spite of therandom initialization which is far away from the true values. In FIG.48, each point on the plot indicates the bearing estimate provided bythe analysis of the block that ends at that point. After just one block,the bearing estimates are very close to true values and port switchingis not observed. The port that captures the AM source undergoes somevariation during the fourth block since during this period the speechsignal has very low power and the bearing estimate based on this blockcannot be accurate. The estimates from the fourth port are not shown,since this port is not activated.

Next, we compare the original speech waveform and its estimate providedby the second port. FIG. 49 illustrates the original speech waveform. Itis important to note the power variation of the speech signal. Thewaveform recovered by port 2 is shown in FIG. 50. It is clear that theeCURE algorithm exhibits rapid convergence for waveform recovery, afterthe second block the waveform estimate closely follows the originalspeech waveform. Many blind signal separation algorithms (e.g., theconstant modulus algorithm) fail to separate signals with high amplitudevariations. In this example, we observe that eigenCURE can also providegood waveform continuity in addition to separation.

In FIG. 51, we show the magnitude of the Port 1 output which capturesthe second FM source that illuminates the array from 0 degree. The idealoutput should be constant since the FM signal is constant modulus. Thevariations decrease as the algorithm adapts, and reach a minimum that isdetermined by the scenario under consideration. Finally, we compare thespectrum of the port 3 output, and the spectrum of the first FM signalwhich it is tracking. We first illustrate the spectrum of the firstsensor measurement in FIG. 52, which shows the contributions from thethree directional signals. FIG. 53 shows the spectrum of port 3 output,which closely resembles to the first FM source it is tracking. Forcomparison, we refer the reader to FIG. 45.

21.9.2 Experiment 2-Transient Sources:

In this experiment, we simulate a case in which there are five farfieldsources and 25,000 snapshots. We use the eCURE algorithm with the sameparameters of the first experiment, the block size is 1000 snapshots.The first source is the DSB-SC AM source from the previous experiment.We scale it so that its power is 30 dB with respect to spatially whitemeasurement noise which is at 0 dB. The second source is a single-toneFM signal at 30 dB, with the following parameters:

f₂=0.04, β₂=0.4, α₂=0.1

The third and fifth signals are Gaussian noise modulated phase modulated(PM) signals which can be characterized as below:

s _(k)(t)=A _(k)exp(j2π(f _(k) t+β _(k) e _(k)(t))), k=3,5

in which {e₃(t), e₅(t)} are independent Gaussian processes with zeromean and unit variance. We have the amplitudes adjusted so that the PMsignals are at 30 dB when they are ON. In addition, we set

f₃=−0.02, β₃=0.2,

f₅=−0.2, β₅=0.2,

The third source is OFF during the interval [6301,15000], and the fifthsource is OFF during the interval [8901,11200]. Therefore, the thirdsource is absent in blocks [8,15], and the fifth source is absent inblocks [10,11]. Finally, the fourth source is a continuous-wave (CW)which turns on after 17,900 snapshots, and can be expressed as:

s₄(t)=A₄exp(j2π(−0.4t)),

This source is also at 30 dB when it is ON. The directions of arrivalfor these five sources are: θ₁=−10°, θ₂=0°, θ₃=5°, θ₄=15°, θ₅=20°. Thefourth source turns ON after all of the four ports are allocated toexisting sources and hence incorporated in this experiment todemonstrate the ability of the eCURE algorithm to eliminate portswitching in the case of new interferers.

In FIG. 54, we show the results from the number of sources estimator. Inthis figure, each point shows the estimated number of sources precedingitself. For example, the second point gives us an estimate of sources inthe snapshots [11001,2000]. For every block, the results are accurate.Source transitions are indicated in the figure.

FIG. 55 shows the bearing estimates obtained from the steering vectorestimates provided by the eCURE algorithm. Similar to FIG. 54, eachpoint for a specific port shows the bearing estimate for the block ofsnapshots preceding itself. For example, the third point for port 1,gives the bearing estimate for the source captured by port 1 during theinterval of [2001,3000] snapshots. The lack of a point (or discontinuityin the graph) indicates port shutdown for that interval during which nobearing estimate is provided. We observe that Port 1 captures Source 1,Port 2 captures Source 2, Port 3 captures Source 3, and Port 4 capturesSource 5. In addition, ports shutdown when the sources they are trackingare turned OFF. It is very important to observe that when the sourcesthat disappeared, turn ON again, they are captured by the same port.Finally, the fourth source cannot cause port switching when it turns ON,all of the four ports continue to track their sources.

To illustrate the waveform recovery performance of the eCURE algorithm,we display the output of port 1 in FIG. 56A and compare it to theoriginal speech signal in FIG. 56B. These figures indicates that eCUREconverges rapidly to extract the speech waveform.

We claim:
 1. A method of complex phase equalization for use in acommunication system using two-dimensional signal constellations, themethod comprising the steps of: receiving and downconverting atwo-dimensional communication signal, to generate in-phase (I) andquadrature (Q) components that have been phase-rotated by an unknownamount during propagation from a transmitter; processing the I and Qsignals in a cumulant recovery (CURE) processing system, to obtain phasecorrection of the I and Q signals prior to signal demodulation; anddemodulating the two-dimensional communication signal after theforegoing processing step, wherein phase correction is effected withoutregard to modulation type and independently of the step of demodulating.2. Complex phase equalizer apparatus for use in a communication systemusing two-dimensional signal constellations, the apparatus comprising: areceiver for receiving a two-dimensional communication signal andgenerating in-phase (l) and quadrature (Q) components, wherein the I andQ components are subject to phase rotation by an unknown angle; acumulant recovery (CURE) processing system connected to receive the Iand Q components from the receiver wherein the I and Q signals receivedby the CURE processing system are each a linear combination of twoindependent source signals, which are the I and Q signals prior to phaserotation, and wherein the CURE processing system derives the twoindependent source signals and effectively compensates for the phaserotation of the I and Q signals; and a demodulator connected to receivephase-corrected I and Q signals from the CURE processing system, whereinthe CURE processing system effects phase correction of the communicationsignals prior to demodulation and independently of the demodulation typeand the demodulation process.